---
abstract: 'The eigenfunctions with eigenvalues $1$ or $-1$ of the transfer operator of Mayer are in bijective correspondence with the eigenfunctions with eigenvalue $1$ of a transfer operator connected to the nearest integer continued fraction algorithm.  This is shown by relating these eigenspaces of these operators to cohomology groups for the modular group with coefficients in certain principal series representations.' 
address:
- '[Mathematisch Instituut](http://www.math.uu.nl), [Universiteit Utrecht](http://www.uu.nl), Utrecht, The Netherlands'
- '[Institut für Theoretische Physik](http://www.dynamik.tu-clausthal.de/), [Technische Universität Clausthal](http://www.tu-clausthal.de), Clausthal-Zellerfeld, Germany'
author:
- 'R.W. Bruggeman'
- 'T. Mühlenbruch'
date: 'July 4, 2007'
title: Eigenfunctions of transfer operators and cohomology
---

[^1]

Introduction {#A}
============

D.Mayer, [@Ma90], defined the operator $$\label{A.1}
{\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}f(z) = \sum_{n=1}^\infty (z+n)^{-2s} \, f\left(
\frac{1}{z+n} \right)$$ on the Banach space of continuous functions on the disk ${\lvertz-1\rvert} \leq \frac{3}{2}$, holomorphic on ${\lvertz-1\rvert} < \frac{3}{2}$, with the supremum norm.  The series converges absolutely if ${\ensuremath{{\mathrm{Re}(s)}}} > \frac{1}{2}$. There is a meromorphic continuation in $s$, with a pole at $\frac12$ as the sole singularity in the region ${\ensuremath{{\mathrm{Re}(s)}}}>0$. The operator ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ is a transfer operator for the Artin billiard dynamical system [@Ar].  It is connected to the Gauss map $x\mapsto \frac1x-[\frac1x]$ on $[0,1]$. Ultimately, this dynamical system comes from closed billiard flows on the quotient of the upper half plane by ${\ensuremath{{\mathrm{PGL}\!\left(2, {\ensuremath{\mathbb{Z}}}\right)}}}$. (Here $\pm {{\begin{pmatrix} -1 & 0 \\ 0 & 1
\end{pmatrix}}}$ acts by $z\mapsto -\bar z$.) An introductory lecture on Mayer’s transfer operator is [@Ma]. 

The DFG research project *Transfer operators and non arithmetic quantum chaos* (Ma 633/16-1) involves finding transfer operators connected to the dynamical systems of closed geodesic flows on the hyperbolic surfaces represented by the quotient of the upper half plane by arbitrary Hecke triangle groups.  For the modular group, it leads to another transfer operator ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$. This operator acts in the space of vectors of two holomorphic functions on the open unit disk which are continuous on the closed unit disk.  With the supremum norm these vectors form a Banach space.  The operator is given by $$\begin{aligned}
\label{A.5}
{\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} \vec f
&=& {{\begin{pmatrix} \tilde {\ensuremath{{\ensuremath{\mathcal{L}}}}}_{s}^{1,1} & \tilde {\ensuremath{{\ensuremath{\mathcal{L}}}}}_{s}^{1,2} \\ \tilde
{\ensuremath{{\ensuremath{\mathcal{L}}}}}_{s}^{2,1} & \tilde {\ensuremath{{\ensuremath{\mathcal{L}}}}}_{s}^{2,2} 
\end{pmatrix}}} \vec{f} \qquad\mbox{with }\\
\nonumber
{\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} ^{1,1} f_1(z)
&=& \sum_{n=3}^\infty (z+n)^{-2s} f_1\left( \frac{-1}{z+n}\right), \\
\nonumber
{\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} ^{1,2} f_2(z)
&=& \sum_{n=2}^\infty (n-z)^{-2s} f_2\left( \frac{1}{-z+n}\right), \\
\nonumber
{\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} ^{2,1} f_1(z)
&=& \sum_{n=2}^\infty (z+n)^{-2s} f_1\left( \frac{-1}{z+n}\right),
\qquad \mbox{and}\\
\nonumber
{\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} ^{2,2} f_2(z)
&=& \sum_{n=3}^\infty (n-z)^{-2s} f_2\left( \frac{1}{-z+n}\right)\end{aligned}$$ This converges absolutely for ${\ensuremath{{\mathrm{Re}(s)}}}>\frac{1}{2}$, and has a meromorphic continuation in $s$ with a first order pole at $s=\frac12$ as sole singularity in the region ${\ensuremath{{\mathrm{Re}(s)}}}>0$.

Our main result is:

\[thm-main\]Let $s\in {\ensuremath{\mathbb{C}}}$, $0<{\ensuremath{{\mathrm{Re}(s)}}}<1$, $s\neq\frac12$. There is a bijective correspondence between the spaces $\ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}-\nobreak1)  \oplus \ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}+\nobreak1)$ and $\ker({\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} - \nobreak1)$.

The eigenfunctions of both transfer operators satisfy finite linear identities.  Lewis and Zagier [@LZ01 Proposition in §3, Chap.  IV] show that if ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}} f=\pm f$, then $P(z)=f(z-\nobreak1)$ satisfies the three term equation $$\label{3te}
P(z) \;=\; P(z+1)+(z+1)^{-2s}P\left( \frac z{z+1}\right)$$ and the parity condition $$z^{-2s}P(1/z) \;=\; \pm P(z)\,.$$ These functions extend holomorphically to ${\ensuremath{\mathbb{C}}}'={\ensuremath{\mathbb{C}}}{\smallsetminus}(-\infty,0]$. So $\ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}-\nobreak1) 
\oplus \ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}+\nobreak1)$ corresponds to a subspace of the space of all holomorphic solutions of on ${\ensuremath{\mathbb{C}}}'$. This subspace is characterized by the asymptotic behavior $P(x) = c_\infty x^{1-2s}+O(x^{-2{\ensuremath{{\mathrm{Re}(s)}}}})$ as $x\uparrow\infty$, and $P(x) = c_0 x^{-1} + O(1)$ as $x\downarrow0$.

For both transfer operators, we relate, in §\[C\], the eigenfunctions on disks to eigenfunctions in the real analytic functions on an interval.  This allows a cohomological interpretation, to be discussed in §\[sect-efMa\] and §\[D\].  We will show that solutions of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} \vec f=\vec f $ correspond to solutions of the four term equation $$\label{4te}
g(z) + (z+2)^{-2s} g\left(\frac{-1}{z+2}\right) \;=\; g(z-1) +
(2-z)^{-2s} g\left( \frac{1-z}{z-2}\right)\,,$$ on a suitable domain containing $(-1,1)$.

Not all solutions of , respectively , correspond to eigenfunctions of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$, respectively eigenfunctions of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$. We will see in Theorem \[thm-th\] and Proposition \[prop-an-coh\] that the space of all real analytic solutions of on the interval $$\left( \frac{-3-\sqrt5}2,\frac{1+\sqrt 5}2\right)$$ is isomorphic to the first cohomology group of the modular group with coefficients in the principal series representation with spectral parameter $s$. Theorem \[thm-par\] shows that the eigenspace of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ for the eigenvalue $1$ corresponds to a well defined subspace of this cohomology group.  This same subspace is also isomorphic to the sum of the eigenspaces of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ for the eigenvalues $1$ and $-1$. This can be shown by methods in [@LZ01] and [@BLZ]. 

We will take care to indicate the various steps in the correspondence between eigenspaces of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ and ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ as explicitly as possible, even for steps where we might refer to [@LZ01] or [@BLZ].  The least explicit step is an application of Proposition \[D2.6\], where a function with two singularities is written as a difference of two functions which each have a singularity in only one point. 

As background information, we discuss in §\[B8\] how ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ arises from the nearest integer continued fraction transformation.  Our proof of the correspondence does not use that both transfer operators arise from the geodesic flow on related quotients of the upper half plane.  It would be interesting to go directly from the geodesic flow to the relevant cohomology groups. 

The Selberg trace formula relates the recurrent points of the geodesic flow to spectral data.  So both transfer operators have a relation to Maass forms. 

Our cohomological approach to the correspondence is based on ideas in [@LZ01] and [@BLZ].  The leading idea in [@BLZ] is the relation between certain cohomology groups and spaces of Maass forms, which we discuss in §\[B7\].  This relates eigenfunctions of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ and ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ to Maass forms without use of the Selberg trace formula. 

We thank R. Sinclair for his remarks on a preliminary version of this paper.  We thank F. Strömberg for the fruitful discussions of the transfer operator and the underlying dynamical system. 

The transfer operator for the nearest integer continuous fraction algorithm {#B8}
---------------------------------------------------------------------------

Although the origin of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ from a dynamical system is not used in this paper, it seems right to explain why ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ deserves to be called a *transfer operator*.  It is derived by the Ruelle transfer operator method applied to a dynamical system based on the nearest integer map and associated continued fractions.  Such nearest integer continued fractions have already been discussed in [@Hu]. 

Consider the interval map $$\label{B8.3}
f_3: \quad I_3 \to I_3 ; \qquad x \mapsto S\,x - \left\lfloor S\,x +
\frac{1}{2}\right\rfloor$$ with $I_3 =\left[ -\frac{1}{2}, \frac{1}{2} \right]$ and $\lfloor x \rfloor$ the element $n\in {\ensuremath{\mathbb{Z}}}$ such that $n\leq x <n+1$ if $x>0$, and $n<x\leq n+1$ if $x\leq0$. The function $f_3$ is closely related to the nearest integer continuous fractions. ( We keep the the subscript $3$ since it is a specialization of an interval map associated to Hecke triangle groups.)  Basically $f_3$ acts as the “left-shift” on the space of configurations $(a_1,a_2,\ldots)$ for the nearest integer continued fraction expansion $$[0;a_1,a_2,\ldots]:= \frac{-1}{a_1 + \frac{-1}{a_2 +
\frac{-1}{\ldots}}} \in I_3\,.$$ The map $f_3$ generates a discrete dynamical system of finite type.  The transfer operator associated to $f_3$ is $$T_s \, f(x) = \sum_{y \in f^{-1}_3 (x)} \left|\frac{{\ensuremath{\mathrm{d}}}f_3^{-1}(x)}{{\ensuremath{\mathrm{d}}}x} \right|^s \; f(y)$$ defined on a suitable function space.  The expression $\frac{{\ensuremath{\mathrm{d}}}f_3^{-1}(x)}{{\ensuremath{\mathrm{d}}}x}$ denotes the derivative of the appropriate invertible branch of $f_3^{-1}$ in $x$. We find ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} = T_s$ on the Banach space on which ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ is defined.  Forthcoming work in the project Ma633/16-1 of the Deutsche Forschungsgemeinschaft will give more details. 

The dynamical system is related to the geodesic flow on the hyperbolic surface ${\ensuremath{{\mathrm{PSL}\!\left(2, {\ensuremath{\mathbb{Z}}}\right)}}}\backslash {\ensuremath{\mathbb{H}}}$. One can show that the Fredholm-determinant $\det \left( 1 - {\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} \right)$ is essentially equal to the Selberg zeta-function $Z(s)$ for the full modular group ${\ensuremath{\Gamma}}$. Here, essentially equal means that $$\label{B8.1}
Z(s) = \frac{\det \left( 1 - {\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} \right)}{\det \left( 1 - {\ensuremath{\mathcal{K}}}_s
\right)}$$ with ${\ensuremath{\mathcal{K}}}_s$ defined on the same space of pairs of function by $$\label{B8.2}{\ensuremath{\mathcal{K}}}_s { g_1 \choose g_2} = {g_1 \big|_{2s} ST^3 \choose g_1 \big|_{2s}
ST^3}.$$ It is known that ${\ensuremath{\mathcal{K}}}_s$ only admits the eigenvalue $1$ if $s \in \{ -n + \frac{\pi ik}{2\ln (r)}; \; 
n \in {\ensuremath{\mathbb{Z}}}_{\geq 0}, k \in {\ensuremath{\mathbb{Z}}}\}$. These values of $s$ are not in the domain under consideration in this note. 

The zeros of $Z(s)$ on the line ${\ensuremath{{\mathrm{Re}(s)}}} = \frac{1}{2}$ correspond to eigenvalues $s(1-\nobreak s)$ of the hyperbolic Laplace operator.  Hence it is not surprising that eigenfunctions of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ with eigenvalue $1$ can be related to cohomology classes that are themselves related to Maass forms.  See §\[B7\]. 

The advantages of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ compared to Mayer’s transfer operator ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ are that its construction is directly related to the geodesic flow and that the same construction works for all Hecke triangle groups.  The disadvantage of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ compared to ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ is its more complicated structure. 

Definitions and results {#B}
=======================

In this preliminary section we define or recall various concepts to be used in this paper.  Among them are the principal series representations in §\[B3\], and the definition of parabolic cohomology groups in §\[B4\].  We state in §\[sect-LM-coh\] and §\[sect-hypcoh\] the results implying the main result Theorem \[thm-main\].  Finally we give in §\[B7\] background information on the relation between certain spaces of Maass forms and cohomology groups. 

Modular group {#B1}
-------------

We use ${{\begin{bmatrix} a & b \\ c & d
\end{bmatrix}}}$ to denote $\left\{ {{\begin{pmatrix} ta & tb \\ tc & td
\end{pmatrix}}} \;:\; t\neq0\right\}$ in $${\ensuremath{{\mathrm{PGL}\!\left(2, {\ensuremath{\mathbb{R}}}\right)}}} = {\ensuremath{{\mathrm{GL}\!\left(2, {\ensuremath{\mathbb{R}}}\right)}}} / \left\{ {{\begin{pmatrix} t & 0 \\ 0 & t
\end{pmatrix}}} \;:\; t\in
{\ensuremath{\mathbb{R}}}{\smallsetminus}\{0\}\right\}\,.$$

We work with the full modular group ${\ensuremath{\Gamma}}={\ensuremath{{\mathrm{PSL}\!\left(2, {\ensuremath{\mathbb{Z}}}\right)}}}$, which is discrete in ${\ensuremath{{\mathrm{PSL}\!\left(2, {\ensuremath{\mathbb{R}}}\right)}}}$, and is generated by $S={{\begin{bmatrix} 0 & -1 \\ 1 & 0
\end{bmatrix}}}$ and $T={{\begin{bmatrix} 1 & 1 \\ 0 & 1
\end{bmatrix}}}$, with the relations $$\label{B1.1}
S^2 = (ST)^3 = 1.$$ We denote also $$\label{B1.3}
{\ensuremath{T^\prime}}: = TST = ST^{-1}S= {{\begin{bmatrix} 1 & 0 \\ 1 & 1
\end{bmatrix}}} \in {\ensuremath{\Gamma}}\quad \mbox{and} \quad C = {{\begin{bmatrix} 0 & 1 \\ 1 & 0
\end{bmatrix}}} \in {\ensuremath{{\mathrm{PGL}\!\left(2, {\ensuremath{\mathbb{Z}}}\right)}}}.$$ We have ${\ensuremath{{\mathrm{PGL}\!\left(2, {\ensuremath{\mathbb{Z}}}\right)}}} = {\ensuremath{\Gamma}}\cup C \, {\ensuremath{\Gamma}}$.

Principal series representations {#B3}
--------------------------------

We describe the standard realization of the principal series representations of ${\ensuremath{{\mathrm{PGL}\!\left(2, {\ensuremath{\mathbb{R}}}\right)}}}$ in the functions on ${\ensuremath{\mathbb{R}}}$.

The group ${\ensuremath{{\mathrm{PGL}\!\left(2, {\ensuremath{\mathbb{R}}}\right)}}}$ has a family of actions, parametrized by $s\in {\ensuremath{\mathbb{C}}}$, on functions defined on a subset of ${\ensuremath{\mathbb{R}}}$, given by $$\label{B2.1a} h|_{2s} M \,(x) = {\lvertad-bc\rvert}^s \,
{\lvertcx+d\rvert}^{-2s} \, h\!\left(\frac{ax+b}{cx+d}\right).$$ This is a right action; the natural place for the symbol $|_{2s}$ is after the function.  We call $s$ the *spectral parameter*. 

For each value of $s$, this action preserves the spaces ${\ensuremath{\mathcal{V}}}^\omega_s$ and ${\ensuremath{\mathcal{V}}}^\infty_s$ of real-analytic and smooth vectors in the discrete series representation with spectral parameter $s$. The space ${\ensuremath{\mathcal{V}}}^\omega_s$ consists of the $h:{\ensuremath{\mathbb{R}}}\rightarrow{\ensuremath{\mathbb{C}}}$ that are real-analytic on ${\ensuremath{\mathbb{R}}}$ and for which $x\mapsto |x|^{-2s}h(-1/x)$ extends as a real-analytic function on a neighborhood of $0$. Replace ‘real-analytic’ by ‘smooth’ to obtain the characterization of $h\in {\ensuremath{\mathcal{V}}}^\infty_s$.

We refer to [@BLZ Section 2] for a discussion of other models of the principal series representations.  Here it suffices to note that elements of ${\ensuremath{\mathcal{V}}}^\omega_s$ and ${\ensuremath{\mathcal{V}}}^\infty_s$ can be viewed as functions on the projective line ${{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}={\ensuremath{\mathbb{R}}}\cup\{\infty\}$, and that the required behavior of $$\label{B3.1}
x\mapsto \left(h|_{2s} S\right)(x) =|x|^{-2s} h(-1/x)$$ may be viewed as the description of analyticity or smoothness at $\infty$.

The real-analytic functions in ${\ensuremath{\mathcal{V}}}^{\omega}_s$ are the restriction of a holomorphic function on some neighborhood of ${\ensuremath{\mathbb{R}}}$, that depends on the functions.  On such holomorphic functions the slash operator takes the form $$\label{B2.1}
h|_{2s} M \,(z) := {\lvertad-bc\rvert}^s \, \big((cz+d)^2\big)^{-s} \,
h\!\left(\frac{az+b}{cz+d}\right).$$ The factor $\big((cz+d)^2\big)^{-s}$ is holomorphic in $z$ for ${\ensuremath{{\mathrm{Re}(z)}}}\neq -\frac dc$. In some cases, for instance for period functions satisfying , we may prefer to choose the factor differently, such that it is holomorphic on the domain of the function and positive on the real points in the domain. 

We need more spaces related to ${\ensuremath{\mathcal{V}}}^\omega_s$ and ${\ensuremath{\mathcal{V}}}^\infty_s$. If $I\subset{{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}$ is an open subset, we define ${\ensuremath{\mathcal{V}}}^\omega_s(I)$ as the space of $h:I\cap{\ensuremath{\mathbb{R}}}\rightarrow {\ensuremath{\mathbb{C}}}$ that are real-analytic on $I\cap {\ensuremath{\mathbb{R}}}$, and for which in the case $\infty\in I$ the function in is real-analytic at $0$. In particular, if $I={{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}{\smallsetminus}E$ for some finite set $E$, then ${\ensuremath{\mathcal{V}}}^\omega_s(I)$ consists of analytic vectors with finitely many singularities on ${{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}$.

The space ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ is defined as the inductive limit $${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s = \lim_{\longrightarrow} {\ensuremath{\mathcal{V}}}^\omega_s({{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}{\smallsetminus}E),$$ where $E$ runs through the finite subsets of ${{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}$. If $h\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$, then there is a minimal finite set $E\subset{{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}$ such that $h\in {\ensuremath{\mathcal{V}}}^{\omega}_s({{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}{\smallsetminus}E)$. We call the elements of $E$ the *singularities* of $h$ and denote this set by ${\ensuremath{\mathrm{Sing}\left({h}\right)}}$.

By imposing conditions at the singularities, we define subspaces of ${\ensuremath{\mathcal{V}}}^{{{\omega}^\ast}}_s$. For instance $${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\infty}}}_s := {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s \cap {\ensuremath{\mathcal{V}}}^\infty_s$$ is the space of smooth vectors that are real-analytic outside finitely many points.  A slightly larger space is ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$ consisting of the $h\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ for which we allow simple pole at finitely many points.  Its elements $f$ have to satisfy at their finitely many singularities $x_0$: $$\begin{aligned}
\label{simple-sing}
&& x \mapsto (x-x_0) \, f(x) \mbox{ is smooth at } x_0 \mbox{ if } x_0
\not= \infty, \mbox{ and}\\
\nonumber
&& y \mapsto y \, {\lverty\rvert}^{-2s} \,f (-1/y) = y \, \left(f|S \right)(y)
\mbox{ is smooth at } 0 \mbox{ if } x_0 = \infty\,.\end{aligned}$$

Thus we have various spaces, all invariant under the action $|_{2s}$ of ${\ensuremath{{\mathrm{PGL}\!\left(2, {\ensuremath{\mathbb{R}}}\right)}}}$, that satisfy the following inclusions: $$\xymatrix{ {\ensuremath{\mathcal{V}}}^{\omega}_s\ar@{^{(}->}@<-.4ex>[r]
& {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\infty}}}_s\ar@{^{(}->}@<-.4ex>[r] \ar@{^{(}->}[d]
& {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s \ar@{^{(}->}@<-.4ex>[r]
& {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s\\
& {\ensuremath{\mathcal{V}}}^\infty_s & }$$

Throughout this note we use the assumption $$\label{B2.2}
0<{\ensuremath{{\mathrm{Re}(s)}}}<1,\qquad s \neq \frac{1}{2}.$$ Mostly, we work with a fixed value of the spectral parameter $s$. Then we shall write $h|M$ instead of $h|_{2s}M$.

For vector valued functions we write ${f_1\choose f_2} |_{2s} M$ for ${f_1 |_{2s} M \choose f_2 |_{2s} M}$. The slash operator extends to elements of the group ring ${\ensuremath{\mathbb{C}}}[{\ensuremath{{\mathrm{PGL}\!\left(2, {\ensuremath{\mathbb{R}}}\right)}}}]$ by $$f|_{2s} (M_1 + a M_2) = f|_{2s} M_1 + a \, f|_{2s} M_2
\qquad\mbox{for all } g_1,g_2 \in {\ensuremath{\Gamma}}\mbox{ and } a \in {\ensuremath{\mathbb{C}}}.$$ In some circumstances one can make sense of $f|_{2s}\Xi$ where $\Xi$ is an infinite sum of elements of ${\ensuremath{\Gamma}}$.

Cohomology groups {#B4}
-----------------

As usual, the *first cohomology group* of ${\ensuremath{\Gamma}}$ with values in a right ${\ensuremath{\Gamma}}$-module $V$ can be described by $$\begin{aligned}
\label{B4.1}
H^1({\ensuremath{\Gamma}}; V)
&=& Z^1({\ensuremath{\Gamma}}; V) / B^1({\ensuremath{\Gamma}}; V),\\
\nonumber
Z^1({\ensuremath{\Gamma}}; V)
&=& \big\{ \psi: {\ensuremath{\Gamma}}\to V; \; \psi_{\gamma \delta} = \psi_\gamma |
\delta + \psi_\delta \mbox{ for all } \gamma,\delta \in {\ensuremath{\Gamma}}\big\}
\quad \mbox{and}\\
\nonumber
B^1({\ensuremath{\Gamma}}; V)
&=& \big\{ \psi \in Z^1({\ensuremath{\Gamma}}; V); \; \exists v \in V \mbox{ such that
} \psi_\gamma = v|(1-\gamma) \big\}.\end{aligned}$$ We give the arguments of the $\emph{cocycles}$ $\psi \in {\ensuremath{\mathbb{Z}}}^1({\ensuremath{\Gamma}};V)$ by a subscript.  Furthermore we denote the right action of ${\ensuremath{\Gamma}}$ on $V$ as $v \mapsto v | \gamma$ for $\gamma \in {\ensuremath{\Gamma}}$ and $v \in V$. If the cohomology group is clear we use the notation $[{\psi}]$ for the cohomology class of the cocycle ${\psi}$.

The *first parabolic cohomology group* is the subgroup of $H^1({\ensuremath{\Gamma}}; V)$ given by $$\begin{aligned}
\label{B4.2}
H_{{\mathrm{par}}}^1({\ensuremath{\Gamma}}; V)
&=& Z_{{\mathrm{par}}}^1({\ensuremath{\Gamma}}; V) / B^1({\ensuremath{\Gamma}}; V),\\
\nonumber
Z_{{\mathrm{par}}}^1({\ensuremath{\Gamma}}; V)
&=& \big\{ \psi \in Z^1({\ensuremath{\Gamma}}; V); \; \exists v \in V \mbox{ such that
} \psi_T = v|(1-T) \big\}.\end{aligned}$$ If $W \supset V$ is a larger ${\ensuremath{\Gamma}}$-module, then the *first mixed parabolic cohomology group* is given by $$\begin{aligned}
\label{B4.3}
H_{{\mathrm{par}}}^1({\ensuremath{\Gamma}}; V,W)
&\!\!\!=\!\!\!&  Z_{{\mathrm{par}}}^1({\ensuremath{\Gamma}}; V,W) / B^1({\ensuremath{\Gamma}}; V),\\
\nonumber
Z_{{\mathrm{par}}}^1({\ensuremath{\Gamma}}; V,W)
&\!\!\!=\!\!\!&  \big\{ \psi \in Z^1({\ensuremath{\Gamma}}; V); \; \exists v \in W
\mbox{ such that } \psi_T = v|(1-T) \big\}.\end{aligned}$$ Note that $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};V,W)$ is a subspace of $H^1({\ensuremath{\Gamma}};V)$, and that there is a natural map $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};V,W) \longrightarrow
H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};W)$.

\[B4.4\] In the definitions above we use that ${\ensuremath{\Gamma}}\backslash {\ensuremath{\mathbb{H}}}$ has only one ${\ensuremath{\Gamma}}$-class of cusps represented by $\infty$, and that the subgroup ${\ensuremath{\Gamma}}_\infty\subset{\ensuremath{\Gamma}}$ fixing $\infty$ is generated by $T$. We refer to [@BLZ Section 10] for a discussion of parabolic cohomology for more general discrete subgroups of ${\ensuremath{{\mathrm{PSL}\!\left(2, {\ensuremath{\mathbb{R}}}\right)}}}$.

Any cocycle is determined by its values on a set of generators, so by ${\psi}_S$ and ${\psi}_T$ for ${\ensuremath{\Gamma}}$. The relations determine the relations $$\label{S-ST-rel}
{\psi}_S |(1+S) \;=\; 0\,,\qquad {\psi}_{T^{-1}S}|\left( 1 + T^{-1}S+
ST\right)\;=\;0\,.$$ For a parabolic cocycle ${\psi}$ we can arrange ${\psi}_T=0$ without changing the cohomology class.  The resulting cocycle is determined by its value on $S$, subject to the relations $$\label{parcocrel}
{\psi}_S|_{2s}(1+S)\;=\;0\,,\qquad {\psi}_S \;=\; {\psi}_S|_{2s}(T+{\ensuremath{T^\prime}})\,.$$

Eigenfunctions of the Mayer operator and parabolic cohomology {#sect-LM-coh}
-------------------------------------------------------------

In the Introduction we have already mentioned that eigenfunctions of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ with eigenvalue $1$ or $-1$ give rise to elements of ${\ensuremath{{}^3\mathrm{FE}}}_s({\ensuremath{\mathbb{C}}}')_{\omega}$, where for $X\subset {\ensuremath{\mathbb{C}}}$: $$\begin{aligned}
\label{FEd-def}
{\ensuremath{{}^3\mathrm{FE}}}_s(X)_{\omega}&\;=\; \bigl\{ \text{analytic }P:X\rightarrow{\ensuremath{\mathbb{C}}}\;:\; P
= P|_{2s}T + P|_{2s}{\ensuremath{T^\prime}}\\
\nonumber
&\qquad \qquad \text{ on } X\cap T^{-1} X \cap (T')^{-1}X\bigr\}\,.\end{aligned}$$ By analytic on $X\subset{\ensuremath{\mathbb{R}}}$ we mean *real analytic*.  For open $X\subset {\ensuremath{\mathbb{C}}}$, analytic means *holomorphic*. 

The cocycle condition ${\psi}_S={\psi}_S|(T+\nobreak{\ensuremath{T^\prime}})$ in is similar to the three term equation .  In [@BLZ Section 13], various aspects of the relation between ${\ensuremath{{}^3\mathrm{FE}}}_s({\ensuremath{\mathbb{C}}}')_{\omega}$ and cohomology are discussed.  For the present paper it is important that under assumption : $$H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s) \;\cong\;
H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s) \;\cong\;
{\ensuremath{{}^3\mathrm{FE}}}_s(0,\infty)_{\omega}^{\mathrm{simple}}\,.$$ The third superscript *simple* indicates that we impose on $P \in {\ensuremath{{}^3\mathrm{FE}}}_s(0,\infty)_{\omega}$ an asymptotic behavior at the end points of $(0,\infty)$: $$\label{as-exp}
P(x) \;\sim\; \sum_{m=-1}^\infty c_m^\infty
x^{-2s-m}\quad(x\uparrow\infty)\,,\qquad P(x) \;\sim\;
\sum_{m=-1}^\infty c_m^0 x^m\quad(x\downarrow0)\,.$$ This is a one-sided version of the behavior at singularities of elements of ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$ defined in §\[B3\]. 

In §\[sect-efMa\] we shall show:

\[prop-efLM-coh\]Let $s\in {\ensuremath{\mathbb{C}}}$, $0<{\ensuremath{{\mathrm{Re}(s)}}}<1$, $s\neq\frac12$. The space $\ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}-\nobreak 1) \oplus \ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}+ \nobreak 1)$ is in bijective correspondence to $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$.

Eigenfunctions of the nearest integer transfer operator and hyperbolic cohomology {#sect-hypcoh}
---------------------------------------------------------------------------------

The definitions and are related to the ${\ensuremath{\Gamma}}$-orbit ${\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}}$ of cusps in ${\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$. The element $T\in {\ensuremath{\Gamma}}$ generates the subgroup ${\ensuremath{\Gamma}}_\infty$ of ${\ensuremath{\Gamma}}$ fixing the element $\infty$ in this orbit.  Let us now work with what we would like to call the *Fibonacci orbit* ${\ensuremath{\mathrm{Fib}}}= {\ensuremath{\Gamma}}(-{\phi})\subset{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$, where ${\phi}=\frac{1+\sqrt 5}2$ is the golden ratio.  The hyperbolic element $TST^2\in {\ensuremath{\Gamma}}$ generates the subgroup ${\ensuremath{\Gamma}}_{-{\phi}}$ of ${\ensuremath{\Gamma}}$ fixing $-{\phi}$.

For ${\ensuremath{\Gamma}}$-modules $W\supset V$: $$\begin{aligned}
\label{hypcoh}
H_{\ensuremath{\mathrm{Fib}}}^1({\ensuremath{\Gamma}}; V,W)
&\!\!\!=\!\!\!&  Z_{\ensuremath{\mathrm{Fib}}}^1({\ensuremath{\Gamma}}; V,W) / B^1({\ensuremath{\Gamma}}; V),\\
\nonumber
Z_{\ensuremath{\mathrm{Fib}}}^1({\ensuremath{\Gamma}}; V,W)
&\!\!\!=\!\!\!&  \big\{ \psi \in Z^1({\ensuremath{\Gamma}}; V): \; \exists v \in W
\;:\; \psi_{TST^2} = v|(1-TST^2) \big\},\end{aligned}$$ and $H^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};V):=H^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};V,V)$.

In particular, $H^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ is a subspace of $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$. The inclusion ${\ensuremath{\mathcal{V}}}^{\omega}_s \hookrightarrow 
{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ induces a linear map $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s) \longrightarrow H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$.

Let us also define for $X\subset {\ensuremath{\mathbb{C}}}$: $$\begin{aligned}
{\ensuremath{{}^4\mathrm{FE}}}_s(X)_{\omega}&\;=\; \biggl\{ \text{analytic }g:X\rightarrow{\ensuremath{\mathbb{C}}}\;:\;\\
\nonumber
&\qquad\qquad \hbox{} g+ g|_{2s}ST^2 \;=\; g|_{2s}T^{-1} +
g|_{2s}T^{-1}ST^{-2}\\
\nonumber
&\qquad\qquad \hbox{} \text{ on }X\cap T^{-2}SX \cap TX \cap T^2STX
\biggr\}\,,\end{aligned}$$ with the same convention concerning analyticity as in §\[sect-LM-coh\].  We shall prove in §\[D2\]:

\[thm-th\]Let $s\in {\ensuremath{\mathbb{C}}}$, $0<{\ensuremath{{\mathrm{Re}(s)}}}<1$, $s\neq\frac12$. There is an injective map ${\vartheta}: {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi}) \longrightarrow
H^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$. The image ${\vartheta}\left( {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi}) \right) \subset
H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ is equal to the image of $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s)$ in $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$.

\[prop-an-coh\]The natural map $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s)\longrightarrow H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ is injective. 

Let ${\psi}\in Z^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s)$ such that ${\psi}_{\gamma}= f|(1-{\gamma})$ for all ${\gamma}\in{\ensuremath{\Gamma}}$ for some $f\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$. From $f|(1-\nobreak T)={\psi}_T\in {\ensuremath{\mathcal{V}}}^{\omega}_s$ it follows that the set of singularities ${\ensuremath{\mathrm{Sing}\left({f}\right)}}$ can contain at most the point $\infty$; otherwise ${\ensuremath{\mathrm{Sing}\left({f}\right)}}$ would be infinite.  Hence ${\ensuremath{\mathrm{Sing}\left({f|S}\right)}} \subset \{0\}$. From $f-f|S ={\psi}_S\in {\ensuremath{\mathcal{V}}}^{\omega}_s$ we conclude that $f$ has no singularities at all, i.e., $f\in {\ensuremath{\mathcal{V}}}^{\omega}_s$. Hence $[{\psi}]=0$ in $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s)$.

We now have the following system of injective maps: $$\xymatrix{ {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi}) \ar@{^{(}->}[r]^{\vartheta}& H^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s) \ar@{^{(}->}[r]
& H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)\\
& H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s) \ar@{^{(}->}[r]
& H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s) \ar@{^{(}->}[u] }$$

\[thm-par\]Let $s\in {\ensuremath{\mathbb{C}}}$, $0<{\ensuremath{{\mathrm{Re}(s)}}}<1$, $s\neq\frac12$. The kernel of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}-1$ determines a subspace of ${\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}$ that is mapped by ${\vartheta}$ onto the image of $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$ in $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$.

This establishes a bijective map between $\ker ({\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}-\nobreak 1)$ and the cohomology group $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$. We prove this theorem in §\[sect-eif-coh\]. 

The results in Proposition \[prop-efLM-coh\], Theorem \[thm-th\], Proposition \[prop-an-coh\] and Theorem \[thm-par\] imply Theorem \[thm-main\].  See also Table \[overview\]. 

$$\xymatrix@R+3ex{ *+[F]{\ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}-1)\oplus \ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}+1)}
  \ar@(dl,ul)@/_/[d]_{\txt{\small \cite[Chap.~IV, \S3]{LZ01},
  Prop.~\ref{prop-Ma-FE}, \ \\\small restriction and extension \ }} \\
*+[F]{{\ensuremath{{}^3\mathrm{FE}}}_s(0,\infty)_{\omega}^{\mathrm{simple}} }
\ar@(dl,ul)@/_/[d]_{\txt{\small Prop.~\ref{prop-FE3-coh}, separation
  of singularities \ }} \ar@(ur,dr)@/_/[u]_{\txt{\small \
  Prop.~\ref{prop-Ma-FE}, bootstrap in\\\small
  \cite[Chap.~III,\S4]{LZ01}}} \\
*+[F]{H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)}
  \ar@{<->}[d]|{\txt{\small Prop.~\ref{prop-an-coh},
  $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s) \rightarrow H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}})$ is
  injective}} \ar@(ur,dr)@/_/[u]_{\txt{\small \
  Prop.~\ref{prop-FE3-coh}, one-sided average ${\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$}} \\
*+[F]{\txt{Image of $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$ in
$H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ }} \ar@(dl,ul)@/_/[d]_{\txt{\small
  Prop.~\ref{prop-thsurj}, one-sided average ${\mathrm{A\!v}}_{TST^2}^+$, \ \\
  \small \S\ref{sect-eif-coh}, one-sided averages}} \\
*+[F]{\txt{subspace of ${\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}$}}
  \ar@(dl,ul)@/_/[d]_{\txt{\small Lemma~\ref{lem-coc-FE}, cocycle
  relations, \ \\\small \S\ref{sect-eif-coh} }}
  \ar@(ur,dr)@/_/[u]_{\txt{\small \ Prop.~\ref{prop-om}, separation of
  singularities, \\\small \S\ref{sect-eif-coh}, one-sided averages}} \\
*+[F]{\txt{solutions of $\vec g|_{2s}{\ensuremath{{\ensuremath{\mathcal{L}}}}}= \vec g$ on intervals} }
  \ar@(dl,ul)@/_/[d]_{\txt{\small Prop.~\ref{prop-bootstrap}, bootstrap
  \ }} \ar@(ur,dr)@/_/[u]_{\txt{\small \ Prop.~\ref{prop-thh}, cocycle
  relations, \\\small \S\ref{sect-eif-coh}}} \\
*+[F]{ \ker({\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}-1) } \ar@(ur,dr)@/_/[u]_{\txt{\small \
  Prop.~\ref{C3.2}, restriction and extension}} }$$

Automorphic forms and cohomology groups {#B7}
---------------------------------------

In this note we work with transfer operators and cohomology groups.  In [@LZ01] and [@BLZ] the main theme is the relation between period functions, automorphic forms and cohomology.  We mention the relevant facts as background material. 

We denote by ${\ensuremath{{}^3\mathrm{FE}}}_s({\ensuremath{\mathbb{C}}}')^0_{\omega}$ the subspace of $P\in {\ensuremath{{}^3\mathrm{FE}}}_s({\ensuremath{\mathbb{C}}}')_{\omega}$, as defined in , with ${\ensuremath{\mathbb{C}}}'={\ensuremath{\mathbb{C}}}{\smallsetminus}(-\infty,0]$, that satisfy $P(x) = O(1)$ as $x\downarrow0$ and $P(x)=O(x^{-2s})$ as $x\rightarrow\infty$. The main theorem in [@LZ01] states that the space ${\ensuremath{{}^3\mathrm{FE}}}_s({\ensuremath{\mathbb{C}}}')_{\omega}^0 \cong H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\infty}}}_s)$ is in bijective correspondence with the space of Maass cusp forms with spectral parameter $s$. A Maass cusp form is a function $u:{\ensuremath{\mathbb{H}}}\longrightarrow{\ensuremath{\mathbb{C}}}$ satisfying $u({\gamma}z) = u(z)$ for all ${\gamma}\in {\ensuremath{\Gamma}}$ that is given by a convergent Fourier expansion $$u(x+iy) \;=\; \sum_{n\neq 0} A_n e^{2\pi i nx} \sqrt y\,
K_{s-1/2}(2\pi|n|y)\,.$$ The space $M_s^0$ of such Maass cusp forms is known to be non-zero only for a discrete set of values of $s$ satisfying ${\ensuremath{{\mathrm{Re}(s)}}}=\frac12$, $s\neq\frac12$.

A slightly larger space of ${\ensuremath{\Gamma}}$-invariant functions is $M_s^1$, consisting of the ${\ensuremath{\Gamma}}$-invariant $u$ on ${\ensuremath{\mathbb{H}}}$ with a converging Fourier expansion $$u(x+iy) \;=\; A_0 y^{1-s} + \sum_{n\neq 0} A_n e^{2\pi i nx} \sqrt y\,
K_{s-1/2}(2\pi|n|y)\,.$$ This space is equal to $M_s^0$ for ${\ensuremath{{\mathrm{Re}(s)}}}=\frac12$, $s\neq \frac12$. For values of $s$ with $0<{\ensuremath{{\mathrm{Re}(s)}}}<\frac12$ such that $\zeta(2s) = 0$ the residue of the Eisenstein series is an element of $M_s^1$. The results in [@BLZ Section 11] show that for $0<{\ensuremath{{\mathrm{Re}(s)}}}<1$, $s\neq\frac12$ there is a bijective correspondence between $M_s^1$ and $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$. These spaces are finite dimensional, and zero for most values of $s$. All elements of $M_s^1$ are eigenfunctions of the hyperbolic Laplace operator: $-y^2\left( \partial_y^2+\partial_x^2\right)u = s
(1-\nobreak s)u$.

The conclusion is that the eigenfunctions of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ with eigenvalues $1$ and $-1$, and the eigenfunctions of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ with eigenvalue $1$ are in bijective correspondence to elements of the space $M_s^1$. This gives a confirmation of the relation between eigenfunctions of transfer operators and automorphic forms that we know already from the relation via the Selberg zeta function. ( See [@Ma] and .) 

The transfer operators on disks and intervals {#C}
=============================================

In the context of dynamical systems one usually considers transfer operators in Banach spaces of holomorphic functions on a disk.  For the relation to cohomology groups with values in principal series spaces, it is more natural to consider the corresponding operators on functions on intervals in ${\ensuremath{\mathbb{R}}}$ or ${\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$. We discuss this relation in §\[sect-efMa\] for the Mayer operator and in §\[sect-ef-nito\] for ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$. In §\[sect-4te\] we derive the four term equation  from the transfer operator ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$.

We start with a discussion of one-sided averages. 

One-sided averages {#C1}
------------------

Both in the definition of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ in and in that of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ in , one recognizes infinite sums of the type $f| {\gamma}T^n$ over infinitely many $n\in {\ensuremath{\mathbb{Z}}}$ for a fixed ${\gamma}\in{\ensuremath{{\mathrm{PGL}\!\left(2, {\ensuremath{\mathbb{Z}}}\right)}}}$. The one-sided averages $$\label{C1.1}
{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}= \sum_{n=0}^\infty T^n \qquad \mbox{and} \qquad {\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}=
-\sum_{n=-\infty}^{-1} T^n$$ play also an important role in [@BLZ].  In this subsection we recall the relevant results. 

Consider a function of the form $f=h|S$, where $h$ is holomorphic on a neighborhood of $0$. Then $$\begin{aligned}
\label{avdef}
{\mathrm{A\!v}}_T^+(f) \;=\; f| {\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}(z) &=& \sum_{n=0}^\infty ((z+n)^2)^{-s} \, h
\left( \frac{-1}{z+n} \right)
\quad\mbox{and} \\
\nonumber
{\mathrm{A\!v}}_T^-(f) \;=\; f| {\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}(z) &=& -\sum_{n=1}^\infty ((z-n)^2)^{-s} \,
h \left( \frac{-1}{z-n} \right)\end{aligned}$$ converge absolutely if ${\ensuremath{{\mathrm{Re}(s)}}}>\frac12$, and define $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ as a holomorphic function on a right half-plane, and $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$ on a left half plane.  If $h(0)=0$, the convergence is absolute for ${\ensuremath{{\mathrm{Re}(s)}}}>0$. Using the Hurwitz zeta function for the contribution of the constant term of $h$ at $0$, we obtain in general a meromorphic continuation, with at most a first order singularity at $s=\frac12$ on ${\ensuremath{{\mathrm{Re}(s)}}}>0$. In this note we will understand $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ and $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$ always in this regularized sense.  We have given two notations in .  With $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ we stress that ${\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ is an element of the completion of the group ring of ${\ensuremath{\Gamma}}$, for which we have made sense of the action on certain functions by regularization.  With ${\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}(f)$ we emphasize that this one-sided average defines an operator on suitable spaces of functions.  In this note we will use $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ and $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$.

These one-sided averages satisfy $$\begin{aligned}
2 \label{avTrel1}
f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}|(1-T) &\;=\; f\,,&\qquad f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}|(1-T) &\;=\; f\,,\\
\label{avTrel2}
f|(1-T)|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}&\;=\; f\,,&\qquad f|(1-T)|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}&\;=\; f\,,\\
\label{avTcom}
f|T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}&\;=\; f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}|T \,,&\qquad f|T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}&\;=\; f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}|T\,,\\
\label{avTsplit}
f + f|T |{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}&\;=\; f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}\,,&\qquad
-f|T^{-1} + f|T^{-1}|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}&\;=\; f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\,,\end{aligned}$$ on suitable right half-planes, respectively left half-planes.  These relations hold trivially in the domain ${\ensuremath{{\mathrm{Re}(s)}}}>\frac12$ of absolute convergence, and survive under meromorphic continuation. 

In particular, we consider these one-sided averages for $f\in{\ensuremath{\mathcal{V}}}^{\omega}(I)$ where $I\subset{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$ is a neighborhood of $\infty$. Then $f$ has the form indicated above.  Let us consider a cyclic interval $I=(a,b)_c$ in ${\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$ containing $\infty$. (This means that $a>b$ in ${\ensuremath{\mathbb{R}}}$ and $(a,b)_c = (a,\infty)
\cup \{\infty\}\cup(-\infty,b)$.) As in [@BLZ Section 3] we have:

\[C1.2\]Let $ I \supset (a,b)_c\ni \infty$. If $f\in
{\ensuremath{\mathcal{V}}}^{\omega}_s\bigl(I\bigr)$, then $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s(a,\infty)$ and is represented by a function holomorphic on a neighborhood of $(a,\infty)$ containing a right half-plane, and $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s (\infty,b+\nobreak1)$ is represented by a function holomorphic on a neighborhood of $(\infty,b+\nobreak1)$ containing a left half-plane. 

There are constants $C_m^\ast$ for $m=-1,0,1,\ldots$ such that $$\begin{aligned}
2 \label{avp-as}
f|_{2s}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}(x) &\;\sim\; \sum_{m=-1}^\infty C_m^\ast
x^{-m-2s}&&(x\uparrow\infty)\,,\\
\label{avm-as}
f|_{2s}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}(x)&\;\sim\; \sum_{m=-1}^\infty C_m^\ast
x^{-m}|x|^{-2s}&\quad&
(x\downarrow-\infty)\,.\end{aligned}$$

In particular, if $f\in {\ensuremath{\mathcal{V}}}^{\omega}_s={\ensuremath{\mathcal{V}}}^{\omega}_s({\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}})$, then $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^\pm}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s({\ensuremath{\mathbb{R}}})$.

The asymptotic behavior in and is related to the singularity behavior in the definition of ${\ensuremath{\mathcal{V}}}^{{{\omega}^\ast\!,\mathrm{simple}}}_s$:

\[8.17\] For $f\in {\ensuremath{\mathcal{V}}}^{\omega}_s $ the following statements are equivalent:

1.   $f\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s|(1-T)$.

2.   $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}=f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$.

If $f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}=f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$, then $f|{\mathrm{A\!v}}^\pm|S(z) \sim \sum_{m=-1}^\infty C_m^\ast x^m$ as $\mp x \downarrow0$. Hence $f\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$.

Conversely, suppose that $f=h|(1-\nobreak T)$ with $h\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$. Then $p_+=h-f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ and $p_-=h-f|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$ satisfy $p_\pm |T=p_\pm$. Moreover, $p_+(x)$ has an asymptotic expansion of the form as $x\uparrow \infty$. Hence the periodic function $p_+$ vanishes.  For $p_-$ let $x\downarrow-\infty$.

\[C1.8\] Suppose that $b,c \in {\ensuremath{\mathcal{V}}}^\omega_s$ satisfy $ b | {\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}+ c | {\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s$. Then $b|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}=b|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$, and $ c|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}=c|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$.

Relations and imply that $p=c|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}- c|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$ satisfies $p|_{2s}T=p$, hence $p$ is a periodic function on ${\ensuremath{\mathbb{R}}}$. Put $a= b | {\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}+ c | {\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$. As $x\uparrow\infty$, the term $c|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ has an asymptotic expansion as in , and $c|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}= a-b|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ also has an expansion of this type. ( For $a$ we know that $a(x) = |x|^{-2s} $(analytic in $-1/x$).)  Hence $p(x) \sim x^{-2s} \left( p_{-1} x + p_0 + \cdots\right)$ as $x\rightarrow\infty$. The periodicity implies that $p$ is bounded, hence $p_{-1}=0$. Next $p(x) = {\ensuremath{\mathcal{O}\left( x^{-2{\ensuremath{{\mathrm{Re}(s)}}}} \right)}}$ implies $p=0$, hence $c|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}=c|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$. Now $c|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^\pm}}\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$, and also $b|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^\pm}}\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$.

We can build one-sided averages for other elements of ${\ensuremath{\Gamma}}$. If $\eta\in {\ensuremath{\Gamma}}$ is hyperbolic, for instance $\eta=TST^2$, then the averages ${\mathrm{A\!v}}_\eta^+=\sum_{n\geq 0}\eta^n$ and ${\mathrm{A\!v}}_\eta^-=-\sum_{n\leq -1}\eta^n$ have the properties corresponding to –, if they converge, [@BLZ Section 7].  If the attracting fixed point ${\omega}(\eta)$ of $\eta$ is in the cyclic interval $I\subset{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$, then $f|_{2s}{\mathrm{A\!v}}_\eta^+$ converges without regularization for ${\ensuremath{{\mathrm{Re}(s)}}}>0$ for all $f\in {\ensuremath{\mathcal{V}}}^{\omega}_s(I)$, and provides us with $f|_{2s}{\mathrm{A\!v}}_\eta^+ \in {\ensuremath{\mathcal{V}}}^{\omega}_s\left( I
{\smallsetminus}\{{\alpha}(\eta)\}\right)$, where ${\alpha}(\eta)$ is the repelling fixed point of $\eta$. In particular, ${\mathrm{A\!v}}_\eta^+ : {\ensuremath{\mathcal{V}}}^{\omega}_s \rightarrow {\ensuremath{\mathcal{V}}}^{\omega}_s\left({\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}{\smallsetminus}\{{\alpha}(\eta)
\}\right)$, and similarly ${\mathrm{A\!v}}_\eta^-:{\ensuremath{\mathcal{V}}}^{\omega}_s \rightarrow
{\ensuremath{\mathcal{V}}}^{\omega}_s\left({\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}{\smallsetminus}\{{\omega}(\eta)\}\right)$.

Eigenfunctions of the Mayer operator {#sect-efMa}
------------------------------------

Eigenfunctions of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ with eigenvalue $\pm1$, or briefly $(\pm1)$-eigenfunctions of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$, can be related to eigenfunctions of a similar operator on ${\ensuremath{\mathcal{V}}}^{\omega}_s(0,\infty)$. This statement is almost contained in the results in [@LZ01].  Nevertheless, we recall the main steps in the proof, since we want to describe the correspondence between $\pm1$-eigenfunctions of ${\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}$ and $1$-eigenfunctions of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ explicitly. 

In this subsection we also indicate how to prove Proposition \[prop-efLM-coh\], on the basis of results in [@BLZ]. 

Under the step $P=f|T$, the function $f\in \ker\left( {\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}\mp1\right)$ corresponds to $P$ holomorphic on $|z-\nobreak 2|< \frac32$, and continuous on $|z-\nobreak 2|\leq \frac 32$ satisfying $P|C{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}= \pm P$, with use of the notation introduced in .  This implies on suitable non-empty domains: $$\begin{aligned}
2 P&\;=\; P|T \pm P|TC \,,&\qquad P|C&\;=\; \pm P\,,\\
P&\;=\; P|T + P|{\ensuremath{T^\prime}}\,.\end{aligned}$$ The last equality is the three term equation .  The proposition in [@LZ01 Chap.  IV, §3] shows that these functions extend to ${\ensuremath{\mathbb{C}}}'={\ensuremath{\mathbb{C}}}{\smallsetminus}(-\infty,0]$ and satisfy $P(x) \sim c_{-1} x^{1-2s} + O(x^{-2{\ensuremath{{\mathrm{Re}(s)}}}})$ as $x\uparrow\infty$ for some $c_{-1}$. Since $P$ satisfies $P|T'{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}= P$ on ${\ensuremath{\mathbb{C}}}'$ and $P|C=\pm P$, we have the asymptotic behavior near both end points of $(0,\infty)$, with $c^0_m = \pm c^\infty_m$. Thus we have $P\in {\ensuremath{{}^3\mathrm{FE}}}_s^\pm({\ensuremath{\mathbb{C}}}')_{\omega}^{\mathrm{simple}}$, where the upper index $\pm$ indicates the $(\pm1)$-eigenspace of $C$ in ${\ensuremath{{}^3\mathrm{FE}}}_s({\ensuremath{\mathbb{C}}}')_{\omega}$, and where the superscript *simple* indicates the subspace satisfying . 

Conversely, starting with $P\in {\ensuremath{{}^3\mathrm{FE}}}_s^\pm({\ensuremath{\mathbb{C}}}')_{\omega}^{\mathrm{simple}}$, we have $P|{\ensuremath{T^\prime}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s\left( (0,-1)_c\right)$, where $(0,-1)_c=(0,\infty)
\cup\{\infty\}\cup(-\infty,0)$ denotes a cyclic interval in ${\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$. Hence $P|{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s(0,\infty)$. By : $$\left( P - P|{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}\right)|(1-T) \;=\; P-P|T - P|{\ensuremath{T^\prime}}\;=\;0\,.$$ The asymptotic behavior of $P - P|{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ near $\infty$ shows that this periodic function vanishes.  So $P$ satisfies $P|{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}=P$, or with use of the parity condition, $P|C{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}=\pm P$. The function $P|{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ is holomorphic on a right half plane.  With the parity condition $P|C=\pm P$ this implies that $P$ is holomorphic on a wedge of the form $|\arg z|<{\varepsilon}$. This suffices as the point of departure for the second stage of the bootstrap procedure in [@LZ01 Chap.  III, §4], which gives a holomorphic extension of $P$ to ${\ensuremath{\mathbb{C}}}'$, still satisfying $P|C{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}= \pm P$. This leads to $f=P|T \in \ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}\mp\nobreak1)$.

Thus, we have an explicit bijective correspondence between the following spaces: $$\begin{aligned}
&\ker \left( {\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}\mp 1\right)\,,\\
& \ker\left(T'{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}-1:{\ensuremath{\mathcal{V}}}^{\omega}_s(0,\infty) \longrightarrow
{\ensuremath{\mathcal{V}}}^{\omega}_s(0,\infty)
\right) \\
\nonumber
&\qquad\qquad\hbox{} \cap \ker \left( C\mp1:{\ensuremath{\mathcal{V}}}^{\omega}_s(0,\infty)
\longrightarrow {\ensuremath{\mathcal{V}}}^{\omega}_s(0,\infty)
\right)\,,
\\
&{\ensuremath{{}^3\mathrm{FE}}}_s^\pm({\ensuremath{\mathbb{C}}}')_{\omega}^{\mathrm{simple}}\,, \text{ and }
{\ensuremath{{}^3\mathrm{FE}}}_s^\pm(0,\infty)_{\omega}^{\mathrm{simple}}\,.\end{aligned}$$ As in [@LZ01 Chap.  I, §3], we have ${\ensuremath{{}^3\mathrm{FE}}}_s({\ensuremath{\mathbb{C}}}')_{\omega}^{\mathrm{simple}} =
{\ensuremath{{}^3\mathrm{FE}}}_s^+({\ensuremath{\mathbb{C}}}')_{\omega}^{\mathrm{simple}} \oplus {\ensuremath{{}^3\mathrm{FE}}}_s^-({\ensuremath{\mathbb{C}}}')_{\omega}^{\mathrm{simple}}$. This gives the following result:

\[prop-Ma-FE\]Let $0<{\ensuremath{{\mathrm{Re}(s)}}}<1$, $s\neq\frac12$. There is an explicit bijection between the following two spaces: $$\begin{aligned}
&\ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}-\nobreak1) \oplus \ker({\ensuremath{{\ensuremath{\mathcal{L}}}^\mathrm{Ma}_{s}}}+\nobreak1)\,,\\
&{\ensuremath{{}^3\mathrm{FE}}}_s(0,\infty)_{\omega}^{\mathrm{simple}}\,.\end{aligned}$$

To complete the proof of Proposition \[prop-efLM-coh\] we have to establish a relation between ${\ensuremath{{}^3\mathrm{FE}}}_s(0,\infty)^{\mathrm{simple}}_{\omega}$ and $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$. The least explicit step in the proof is provided by the following result:

\[D2.6\] If $f \in {\ensuremath{\mathcal{V}}}^{{{\omega}^\ast}}_s$ satisfies ${\ensuremath{\mathrm{Sing}\left({f}\right)}} \subset \{\xi,\eta\}$ for two different points $\xi$ and $ \eta$ in $ {\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$, then there are $f_\xi, f_\eta \in {\ensuremath{\mathcal{V}}}^{{{\omega}^\ast}}_s$ such that $f = f_\eta- f_\xi$ and ${\ensuremath{\mathrm{Sing}\left({f_\xi}\right)}} \subset \{\xi\}$, ${\ensuremath{\mathrm{Sing}\left({f_\eta}\right)}} \subset \{\eta\}$. The functions $f_\xi$ and $f_\eta$ are not unique.  The freedom consists of adding the same element of ${\ensuremath{\mathcal{V}}}^\omega_s$ to both functions. 

This follows from, e.g., [@Ho Theorem 1.4.5].  See [@BLZ Section 12] for the application to elements of ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$.

The idea is to use another model of the principal series, in which the elements of ${\ensuremath{\mathcal{V}}}^{\omega}_s$ correspond to functions holomorphic on an annulus in ${\ensuremath{\mathbb{C}}}$ containing the unit circle.  The function $f$ in the proposition is represented by a holomorphic function on an open set ${\Omega}\subset{\ensuremath{\mathbb{C}}}$ containing the unit circle minus the points $\tilde{\xi}$ and $\tilde\eta$ corresponding to ${\xi}$ and $\eta$. Write ${\Omega}= {\Omega}_1\cap {\Omega}_2$ with $\tilde \eta\in {\Omega}_1$, $\tilde {\xi}\in {\Omega}_2$. Apply [@Ho Theorem 1.4.5] with $g_{1,2} = f$ to obtain $f=g_1-g_2$ on ${\Omega}$ with $g_j$ holomorphic on ${\Omega}_j$.

\[prop-FE3-coh\]Let $0<{\ensuremath{{\mathrm{Re}(s)}}}<1$, $s\neq \frac12$. There is an explicit bijection between ${\ensuremath{{}^3\mathrm{FE}}}_s(0,\infty)^{\mathrm{simple}}_{\omega}$ and $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$.

Suppose that $P\in {\ensuremath{{}^3\mathrm{FE}}}_s(0,\infty)^{\mathrm{simple}}_{\omega}$. We extend it to $\tilde P\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ by $$\tilde P\;=\;P\text{ on }(0,\infty)\,,\qquad \tilde
P\;=\;-P|S\text{ on }(-\infty,0)\,.$$ So ${\ensuremath{\mathrm{Sing}\left({\tilde P}\right)}} \subset \{0,\infty\}$. By separate computations on $(-\infty,-1)$ and $(-1,0)$ we conclude that $\tilde P$ satisfies on ${\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}{\smallsetminus}\{\infty,-1,0\}$. Thus, we obtain a cocycle ${\psi}\in Z^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$, determined by $${\psi}_T \;=\;0\,,\qquad {\psi}_S\;=\; \tilde P\,.$$

Since $\tilde P|{\ensuremath{T^\prime}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s\left({\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}{\smallsetminus}\{-1,0\}\right)$, we have $\tilde P |{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}= P |{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s(0,\infty)$ and $\tilde P|{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s(-\infty,0)$. We have indicated earlier in this subsection that the asymptotic behavior of $P(x)$ as $x\uparrow\infty$ implies that $\tilde P|{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}= \tilde P$ on $(0,\infty)$. The asymptotic behavior of $P(x)$ as $x\downarrow 0$ implies a similar asymptotic behavior of $\tilde P(x)$ as $x\downarrow-\infty$, which in turn implies analogously that $\tilde P |{\ensuremath{T^\prime}}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}= \tilde P$ on $(-\infty,0)$. Consulting and we conclude that $\tilde P$ has the same coefficients in its expansions for both directions of approach to $\infty\in {\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$. The fact that $\tilde P|S=-\tilde P$ implies the same statement at $0$. Hence $\tilde P \in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$ and ${\psi}\in Z^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$. It is the unique cocycle in its cohomology class in $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$, since $({\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)^T=\{0\}$, as is shown in [@BLZ Section 8]. 

Proposition \[D2.6\] implies that there are $F_\infty, F_0\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ such that $\tilde P=F_\infty-F_0$ with ${\ensuremath{\mathrm{Sing}\left({F_\infty}\right)}}\subset\{\infty\}$, ${\ensuremath{\mathrm{Sing}\left({F_0}\right)}}\subset\{0\}$. Since $F_\infty$ has the same type of asymptotic behavior at $\infty$ as $\tilde P$, we conclude that $F_\infty, F_0\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$. Moreover, $\tilde P|S=-\tilde P$ implies that $F_0 = F_\infty+{\alpha}$ for some ${\alpha}\in {\ensuremath{\mathcal{V}}}^{\omega}_s$. From ${\ensuremath{\mathrm{Sing}\left({F_\infty|{\ensuremath{T^\prime}}}\right)}}\subset
\{-1\}$ and ${\ensuremath{\mathrm{Sing}\left({F_0|S{\ensuremath{T^\prime}}}\right)}}\subset\{0\}$, it follows that $$\begin{aligned}
{\ensuremath{\mathrm{Sing}\left({\tilde P|(1-T) }\right)}}&\;=\; {\ensuremath{\mathrm{Sing}\left({ F_\infty|{\ensuremath{T^\prime}}-F_\infty|S{\ensuremath{T^\prime}}-{\alpha}|{\ensuremath{T^\prime}}}\right)}}\; \subset\; \{-1,0\}\,,\\
{\ensuremath{\mathrm{Sing}\left({ F_\infty|(1-T)}\right)}} &\;=\;{\ensuremath{\mathrm{Sing}\left({ \tilde P|(1-T) + F_0|(1-T)}\right)}}
\;\subset\; \{-1,0\}\,.\end{aligned}$$ On the other hand ${\ensuremath{\mathrm{Sing}\left({F_\infty}\right)}}\subset\{\infty\}$ implies that ${\ensuremath{\mathrm{Sing}\left({F_\infty|(1-T)}\right)}}\subset\{\infty\}$. The conclusion is that $F_\infty|(1-T)
\in {\ensuremath{\mathcal{V}}}^{\omega}_s$. We conclude that the cocycle $$\tilde{\psi}: {\gamma}\mapsto {\psi}_{\gamma}-F_\infty|(1-{\gamma})$$ takes values in ${\ensuremath{\mathcal{V}}}^{\omega}_s$. The freedom in the choice of $F_\infty$ and $F_0$ amounts to the freedom of choosing $\tilde{\psi}$ in its class in $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$.

Conversely, we start with a cocycle $\tilde {\psi}\in
Z^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$. By the definition of mixed parabolic cohomology in , there exists $v\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$ such that $\tilde {\psi}_T = v|(1-\nobreak T)$. A possible choice is $\tilde{\psi}_T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$, which coincides with $\tilde{\psi}_T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$ according to Lemma \[8.17\], since ${\psi}_T\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$. Lemma \[C1.8\] shows that $v=\tilde{\psi}_T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ is the sole possibility. 

Now ${\psi}_{\gamma}= \tilde{\psi}_{\gamma}- \tilde{\psi}_T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}|(1-\nobreak{\gamma})$ determines the unique cocycle in the cohomology class of $\tilde{\psi}$ in $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$ vanishing on $T$. Note that ${\psi}$ does not change if we change $\tilde{\psi}$ in its cohomology class in $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$. We have ${\psi}_T=0$ and ${\psi}_S = \tilde{\psi}_S -\tilde{\psi}_T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}|(1-\nobreak S) \in
{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$ with singularities contained in $\{0,\infty\}$. Restriction of ${\psi}_S$ to $(0,\infty)$ gives an element of ${\ensuremath{{}^3\mathrm{FE}}}_s(0,\infty)^{\mathrm{simple}}_s$.

Noting that $F_\infty = - \tilde{\psi}_T |{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$, we check that both procedures are inverse to each other. 

There might be cohomology classes in $H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$ that are not represented by a cocycle ${\psi}$ such that ${\ensuremath{\mathrm{Sing}\left({{\psi}_S}\right)}}\subset \{0,\infty\}$. In [@BLZ Section 12] it takes work to show that such classes do not exist. 

Eigenfunctions of the nearest integer transfer operator {#sect-ef-nito}
-------------------------------------------------------

In this subsection we relate $1$-eigenfunctions of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ in to vectors of real analytic functions on intervals.  We note that ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ is given by $$\label{LL-f}
\vec g \mapsto \vec g| {\ensuremath{{\ensuremath{\mathcal{L}}}}}\,,\quad\text{ where } \quad {\ensuremath{{\ensuremath{\mathcal{L}}}}}\;=\;
{{\begin{pmatrix} ST^3{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}& ST^2{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}\\ -ST^{-1}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}& -ST^{-2}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\end{pmatrix}}}\,,$$ for $\vec g=(g_1,g_2)$ continuous on $D=\left\{z\in {\ensuremath{\mathbb{C}}}\;:\; |z|\leq 1\right\}$ and holomorphic on the interior $\mathring D$. The transition from the operator notation in to the right module notation here causes a transition from column vectors to row vectors and a transposition of the matrix. 

For the first component $g_1$ of $\vec g =(g_1,g_2)$ in the domain of ${\ensuremath{{\ensuremath{\mathcal{L}}}}}$ the translate $g_1|ST^3$ is holomorphic on the region $|z+\nobreak 3|> 1$ in ${\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{C}}}}}}$, hence $g_1|ST^3{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}$ is at least defined as a holomorphic function on the right half plane ${\ensuremath{{\mathrm{Re}(z)}}}>-2$ in ${\ensuremath{\mathbb{C}}}$. Proceeding similarly with the other components we find that $\vec g|{\ensuremath{{\ensuremath{\mathcal{L}}}}}$ is holomorphic on $-1<{\ensuremath{{\mathrm{Re}(z)}}}<1$, which contains the interior of the unit disk $D$. Considering two terms in the infinite sum separately, with , we see that $\vec g|{\ensuremath{{\ensuremath{\mathcal{L}}}}}$ is continuous on the unit disk $D$, its boundary included.  Restriction of $\vec g$ to a neighborhood of $(-1,1)$ provides us with a solution of $\vec g|{\ensuremath{{\ensuremath{\mathcal{L}}}}}= \vec g$ in the vector valued analytic functions on $(-1,1)$.

We shall see that the components of this restriction to $(-1,1)$ can be extended to a larger interval, and that conversely solutions of $\vec g = \vec g|{\ensuremath{{\ensuremath{\mathcal{L}}}}}$ on the resulting interval come from $1$-eigenfunctions of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$.

It turns out that a crucial role is played by the intervals $(-{\phi}^2,{\phi})$ and $(-{\phi},{\phi}^2)$, where ${\phi}$ denotes the golden ratio ${\phi}=\frac{1+\sqrt 5}2$.

\[C3.2\] Let $a_1,b_2 \in \left( {\phi}^{-2},{\phi}^2 \right)$ and $a_2,b_1\in (0,1]$. Suppose that $f_1\in {\ensuremath{\mathcal{V}}}^{\omega}_s(-a_1,b_1)$ and $f_2\in {\ensuremath{\mathcal{V}}}^{\omega}_s(-a_2,b_2)$ satisfy $\left( f_1,\, f_2\right)|_{2s}{\ensuremath{{\ensuremath{\mathcal{L}}}}}= \left( f_1,\, f_2 \right)$. Then $f_1 $ is the restriction of $h_1 \in {\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi}^2,{\phi})$ and $f_2$ of $h_2\in {\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi},{\phi}^2)$ such that $(
h_1, h_2)|{\ensuremath{{\ensuremath{\mathcal{L}}}}}= ( h_1, h_2)$, and $h_1=h_2|T$. The values of $h_1(x)$, respectively $h_2(x)$ for each given $x\in
(-{\phi}^2,{\phi})$, respectively $x\in (-{\phi},{\phi}^2)$, can be expressed in values of $f_1$ and $f_2$ by a finite number of applications of the relation $(f_1,f_2)=(f_1,f_2)|_{2s}{\ensuremath{{\ensuremath{\mathcal{L}}}}}$.

By analyticity the eigenfunction equation extends from given open intervals to larger ones, and relation implies $h_1=h_2|T$. The statement that requires work is the extension of the domains. 

Denote $(f_1,f_2)|{\ensuremath{{\ensuremath{\mathcal{L}}}}}$ by $(\tilde f_1, \tilde f_2) $.  We have $$\begin{aligned}
2 f_1|ST^3 &\;\in\; {\ensuremath{\mathcal{V}}}^{\omega}_s\bigl(
({{\textstyle \frac{1}{a_1}}}-3,{{\textstyle \frac{-1}{b_1}}}-3)_c\bigr)\,,&\quad
 f_2|ST^{-1}&\;\in\;{\ensuremath{\mathcal{V}}}^{\omega}_s \bigl(
 ({{\textstyle \frac{1}{a_2}}}+1,{{\textstyle \frac{-1}{b_2}}}+1)_c\bigr)\,,\\
f_1|ST^2 &\;\in\; {\ensuremath{\mathcal{V}}}^{\omega}_s\bigl(
({{\textstyle \frac{1}{a_1}}}-2,{{\textstyle \frac{-1}{b_1}}}-2)_c\bigr)\,,&\quad
 f_2|ST^{-2}&\;\in\;{\ensuremath{\mathcal{V}}}^{\omega}_s \bigl(
 ({{\textstyle \frac{1}{a_2}}}+2,{{\textstyle \frac{-1}{b_2}}}+2)_c\bigr)\,,\end{aligned}$$ where $(x,y)_c=(x,\infty)\cup\{\infty\}\cup(-\infty,y)$ is the notation for cyclic intervals in ${\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}$. With Lemma \[C1.2\]: $$\begin{aligned}
2 f_1|ST^3{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}&\;\in\; {\ensuremath{\mathcal{V}}}^{\omega}_s\bigl(
{{\textstyle \frac{1}{a_1}}}-3,\infty\bigr)\,,&\quad f_2|ST^{-1}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}&\;\in\;
{\ensuremath{\mathcal{V}}}^{\omega}_s\bigl( -\infty, {{\textstyle \frac{-1}{b_2}}}+2\bigr)\,,\\
f_1|ST^2{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}&\;\in\; {\ensuremath{\mathcal{V}}}^{\omega}_s\bigl(
{{\textstyle \frac{1}{a_1}}}-2,\infty\bigr)\,,&\quad f_2|ST^{-2}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}&\;\in\;
{\ensuremath{\mathcal{V}}}^{\omega}_s\bigl( -\infty, {{\textstyle \frac{-1}{b_2}}}+3\bigr)\,,\end{aligned}$$ which implies $$\begin{aligned}
2 \tilde f_1 \;\in\; {\ensuremath{\mathcal{V}}}^{\omega}_s&\bigl(
{{\textstyle \frac{1}{a_1}}}-3, {{\textstyle \frac{-1}{b_2}}}+2\bigr)\,,&\quad \tilde f_2
\;\in\; {\ensuremath{\mathcal{V}}}^{\omega}_s&\bigl( {{\textstyle \frac{1}{a_1}}}-2,
{{\textstyle \frac{-1}{b_2}}}+3\bigr)\,.\end{aligned}$$ The end points of the domains are transformed according to $-\tilde
a_1=\frac1{a_1}-3$, $\tilde b_1 = \tilde b_2-1$, $-\tilde a_2= -\tilde a_1+1$, $\tilde b_2 = 3-\frac1{b_2}$. Iterating this, the $a_1$ starting in $\left( {\phi}^{-2},{\phi}^2\right)$ form a sequence increasing to ${\phi}^2$, and similarly for the $b_2$. This leads to the extension indicated in the proposition. 

This proposition shows that $1$-eigenfunctions of ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ extend to vectors of the form $(g,g|T)$ with $g$ holomorphic on a neighborhood of $(-{\phi}^2,{\phi})$, such that the relation $(g,g|T) |{\ensuremath{{\ensuremath{\mathcal{L}}}}}= (g,g|T)$ is valid on a neighborhood of $(-{\phi},{\phi})$.

\[prop-bootstrap\]Suppose that $g \in {\ensuremath{\mathcal{V}}}^{\omega}_s$ satisfies $$\label{1ef}
(g, g|T^{-1})|{\ensuremath{{\ensuremath{\mathcal{L}}}}}= (g,g|T^{-1})$$ on $(-{\phi},{\phi})$. Then $g$ extends holomorphically to a neighborhood of the closed unit disk $D$ and holds on that neighborhood. 

Since $g$ is real analytic on the interval $(-{\phi}^2,{\phi})
\supset [-1,1]$, there is a complex ${\varepsilon}$-neighborhood $U$ of $[-1,1]$ to which $g$ extends as a holomorphic function.  Relation stays valid on this neighborhood. 

Denote $\vec g = (g,g|T^{-1})$. We have $$\label{A-eq}
\vec g|{\ensuremath{{\ensuremath{\mathcal{L}}}}}\;=\; \sum_{n\geq 0} \vec g | A_n \,,\qquad A_n \;=\;
{{\begin{pmatrix} ST^{3+n} & ST^{2+n} \\ ST^{-2-n} & ST^{-3-n}
\end{pmatrix}}}\,.$$ For sufficiently large $n$ the four images $(A_n)_{i,j} D$, $i,j\in \{1,2\}$, are contained in the given neighborhood $U$ of $[-1,1]$. Lemma \[C1.2\] and repeated application of relation show that there is a tail of the series in representing a holomorphic function on a neighborhood of $D$. Thus we have on an open neighborhood ${\Omega}$ of $(-{\phi},{\phi})$: $$\vec g|{\ensuremath{{\ensuremath{\mathcal{L}}}}}\;=\; \sum_{n=0}^N g|A_n + \left( \text{holomorphic on a
neighborhood of $D$}\right)\,.$$

For $g$ in the remaining terms $g|A_n$ we substitute again.  Repeating this process, we obtain for each $k\geq 1$ on an open neighborhood ${\Omega}_k$ of $(-{\phi},{\phi})$: $$\begin{aligned}
\label{L-iter}
 \vec g(z) &\;=\; \sum_{n_1=0}^{N_1} \cdots \sum_{n_k=0}^{N_k} g|
A_{n_k}\cdots A_{n_1}(z)\\
\nonumber
&\qquad\hbox{}
 + \left( \text{holomorphic on a neighborhood of $D$}\right)\,.\end{aligned}$$ The neighborhood ${\Omega}_k$ is increasing in $k$.

The matrix elements of $A_{n_k}\cdots A_{n_1}$ are of the form $ST^{a_k}ST^{a_{k-1} }\cdots S T^{a_1}$, where $a_j\in {\ensuremath{\mathbb{Z}}}$, $|a_j|\geq 2$, and $a_ja_{j+1}=-4$ if $|a_j|=|a_{j+1}|=2$. (The transfer operator ${\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}}$ is designed to reflect this condition on the $a_j$ in nearest integer continuous fraction expansions.)  Each $ST^{a_j}$ maps the unit disk $ D$ into itself.  For the imaginary part $y$ of $z\in D$ we have $${\ensuremath{{\mathrm{Im}(S T^{a_j} z)}}} \;=\; \frac y{|z+a_j|^2} \;\leq \; \frac
y{(|a_j|-1)^2}\,.$$ So each $ST^{a_j}$ with $|a_j|\geq 3$ decreases the imaginary part with at least a factor $4$. If $a_j=\pm 2$ the imaginary part does not increase and if moreover $j<k$, we know that $a_{j+1}\neq a_j$. One checks for $a_j a_{j+1}\geq 6$ and $a_j a_{j+1} \leq -4$ separately that the imaginary part decreases at least with a factor $4$ under $ST^{a_{j+1}}ST^{a_j}$. So if $k$ is sufficiently large, all $ST^{a_k}\cdots ST^{a_1}z$ with $z\in D$ are contained in the ${\varepsilon}$-neighborhood $U$ of $[-1,1]$ we started with.  For such $k$, all explicit terms in can be absorbed in the last term. 

These propositions imply:

Let $0<{\ensuremath{{\mathrm{Re}(s)}}}<1$, $s\neq\frac12$. Restriction and analytic extension give a bijective correspondence between $\ker({\ensuremath{{\tilde{\ensuremath{\mathcal{L}}}}_{s}}} - \nobreak1)$ and the space of solutions in ${\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi}^2,{\phi})
\times {\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi},{\phi}^2)$ of $$(g_1,g_2)|_{2s} {\ensuremath{{\ensuremath{\mathcal{L}}}}}\;=\; (g_1,g_2)\,.$$

Four term equation {#sect-4te}
------------------

\[prop-1ei-4te\]The solutions in ${\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi}^2,{\phi}) \times {\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi},{\phi}^2)$ of $\vec g = \vec g |_{2s}{\ensuremath{{\ensuremath{\mathcal{L}}}}}$ are of the form $(g,g|T^{-1})$ with $g\in {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}$.

The eigenfunction relations $$\begin{aligned}
2 h_1 &\;=\; & h_1|ST^3{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}&-h_2|ST^{-1}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\,,\\
h_2&\;=\; &h_1|ST^2{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}&-h_2|ST^{-2}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\,,\end{aligned}$$ imply $h_2=h_1|T^{-1}$. So they are equivalent to $$\label{eif1}
h \;=\; h|ST^3 {\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}- h|T^{-1}ST^{-1}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\qquad \text{ on
}(-{\phi}^2,{\phi})\,.$$ Applying $|(1-\nobreak T)$ to , and using , we get $$h|(1-T) \;=\; h|(ST^3 - T^{-1}ST^{-1})\,,$$ as an identity in ${\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi}^2,{\phi}^{-1})$. Apply $|_{2s}T^{-1}$ to obtain as an equality in ${\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi},{\phi})$ $$\label{4te|}
h|(1+ST^2) \;=\; h|(T^{-1}+T^{-1}ST^{-2})\,,$$ which is . 

Four term equation and cohomology {#D}
=================================

In the case of the three term equation the step from solutions on $(0,\infty)$ to cohomology has been discussed in §\[sect-efMa\].  In the case of the four term equation , the situation is more complicated, but the approach is essentially based on the same ideas.  The main steps are indicated in §\[sect-hypcoh\].  In this section we prove Theorems \[thm-th\] and \[thm-par\]. 

A model for parabolic and hyperbolic cohomology {#D1}
-----------------------------------------------

Instead of the inhomogeneous cocycles used in §\[B4\] to describe the first cohomology group, one can also employ homogeneous ones.  To an inhomogeneous cocycle $\psi$ corresponds the *homogeneous cocycle* $\tilde{c}: {\ensuremath{\Gamma}}^2\longrightarrow V$ given by $$\label{D1.1}
\tilde c_{\gamma,\delta} = \psi_{\gamma\delta^{-1}}|\delta
\qquad({\gamma},\delta\in {\ensuremath{\Gamma}})\,.$$ satisfying for ${\gamma},\delta,{\varepsilon}\in {\ensuremath{\Gamma}}$ $$\label{D1.2}
\tilde c_{\gamma{\ensuremath{\varepsilon}},\delta{\ensuremath{\varepsilon}}} = \tilde c_{\gamma,\delta}|{\ensuremath{\varepsilon}}\quad \mbox{and} \quad \tilde c_{\gamma,\delta}+\tilde c_{\delta,{\ensuremath{\varepsilon}}}
= \tilde c_{\gamma,{\ensuremath{\varepsilon}}}.$$ (This implies that $\tilde c_{\gamma,\gamma}=0$ and $\tilde c_{\delta,\gamma}=-\tilde c_{\gamma,\delta}$.) Coboundaries in $B^1({\ensuremath{\Gamma}};V)$ correspond to functions $$\label{D1.3}
(\gamma,\delta)\mapsto v|\delta-v|\gamma$$ with $v\in V$.

Homogeneous cocycles are parabolic if and only if by adding a coboundary, we can arrange $\tilde c_{T,1}=0$.

A homogeneous parabolic cocycle satisfies $$\begin{aligned}
\label{D1.4}
\tilde c_{T\gamma,\delta} &=& \tilde c_{T\gamma,\gamma}+\tilde
c_{\gamma,\delta} = 0|\gamma+\tilde c_{\gamma,\delta} \quad
\mbox{and} \\
\tilde c_{\gamma,T\delta} &=& \tilde c_{\gamma,\delta}+ \tilde
c_{1,T}|\delta = \tilde c_{\gamma,\delta}-0|\delta.\end{aligned}$$ So $\tilde c$ induces a function $(\Gamma_\infty\backslash {\ensuremath{\Gamma}}) ^2 \longrightarrow V$, where $\Gamma_\infty$ is the subgroup of ${\ensuremath{\Gamma}}$ generated by $T$, which is the subgroup fixing $\infty$. Conversely, every such function satisfying induces a homogeneous parabolic cocycle.  Taking into account that the set $\Gamma_\infty\backslash{\ensuremath{\Gamma}}$ can be identified with the set of cusps ${\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}} ={\ensuremath{\mathbb{Q}}}\cup\infty \subset {{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}$ via $\gamma\mapsto \gamma^{-1}\infty$, we obtain the description of $H^1_{\mathrm {par}} ({\ensuremath{\Gamma}};V)$ as the quotient $Z^1_{{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}}}({\ensuremath{\Gamma}};V) / B^1_{{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}}}({\ensuremath{\Gamma}};V)$, where $Z^1_{{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}}}({\ensuremath{\Gamma}};V)$ is the space of maps $c:{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}}\times{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}}\longrightarrow V$ such that $c_{\xi,\eta}+c_{\eta,\zeta} = c_{\xi,\zeta}$ and $c_{\gamma^{-1}\xi,\gamma^{-1}\eta} = c_{\xi,\eta}|\gamma$ for $\gamma\in {\ensuremath{\Gamma}}$ and $\xi,\eta,\zeta\in {\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}}$, and where $B^1_{{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}}}({\ensuremath{\Gamma}};V)$ consists of the subset of elements of the form $c_{\xi,\eta} = f_\eta-f_\gamma$ with $f:{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{Q}}}}}}\longrightarrow V$ satisfying $f_{\gamma^{-1}\xi} = f_\xi|\gamma$.

If we take another base point $\xi\in {{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}$, we can work with cocycles of the same type on other ${\ensuremath{\Gamma}}$-orbits in ${{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}$. If the subgroup ${\ensuremath{\Gamma}}_{\xi}$ of ${\ensuremath{\Gamma}}$ leaving $\xi$ fixed is trivial, we get a description of $H^1({\ensuremath{\Gamma}};V)$. In fact a homogeneous cocycle $\tilde{c}$ on ${\ensuremath{\Gamma}}^2$ corresponds to a cocycle $c$ on ${\ensuremath{\Gamma}}\xi\subset{{\ensuremath{\mathbb{P}^1_{{\ensuremath{\mathbb{R}}}}}}}$ by $$\label{D1.6}
\tilde c_{\gamma,\delta} = c_{\gamma^{-1}\xi,\delta^{-1}\xi}.$$

The situation is different if $\xi$ is a hyperbolic fixed point of ${\ensuremath{\Gamma}}$. Then the procedure indicated above leads to the hyperbolic cohomology group discussed in §\[sect-hypcoh\].  A corresponding homogeneous group cocycle ${\psi}: {\gamma}\mapsto {\psi}_{\gamma}=c_{{\gamma}^{-1}{\xi},{\xi}}$ satisfies $$\psi_H \in V|(1-H)$$ for a generator $H$ of the stabilizer ${\ensuremath{\Gamma}}_\xi$ of the point $\xi$. We will apply this for the stabilizer ${\ensuremath{\Gamma}}_{-{\phi}}$ for the golden ratio ${\phi}=\frac{1+\sqrt 5}{2}$. A generator of ${\ensuremath{\Gamma}}_{-{\phi}}$ is $TST^2$. Thus, we have a model of $H^1_{\ensuremath{\mathrm{Fib}}}$. A group cocycle corresponding ${\psi}$ to the cocycle $c$ on ${\ensuremath{\mathrm{Fib}}}$ is given by ${\psi}_{\gamma}=c_{{\gamma}^{-1}(-{\phi}),-{\phi}}$. In particular, $ c$ is determined by $ c_{-{\phi},{\phi}^{-1}} = - {\psi}_S$ and $ c_{-{\phi},{\phi}} = - {\psi}_{T^{-1}S}$, subject to the relations $$\label{hyprel}
 c_{-{\phi},{\phi}^{-1}}|(1+S)\;=\;0\,,\qquad
 c_{-{\phi},{\phi}}|(1+T^{-1}S+ST)\;=\;0\,;$$ see .  Such a cocycle is a coboundary if there is $v\in V$ such that $v|TST^2=v$ and $c_{{\gamma}^{-1}(-{\phi}),{\delta}^{-1}(-{\phi})} = v|{\delta}-v|{\gamma}$ for ${\gamma},{\delta}\in {\ensuremath{\Gamma}}$.

Solutions of the four term equation and hyperbolic cohomology {#D2}
-------------------------------------------------------------

In this subsection we prove Theorem \[thm-th\].  We work with cocycles on ${\ensuremath{\mathrm{Fib}}}={\ensuremath{\Gamma}}(-{\phi})$ as the model of hyperbolic cohomology.  For the computations we found the graph in Figure \[H-pict\] useful. 

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\[lem-coc-FE\]Suppose that $c$ is a $1$-cocycle on ${\ensuremath{\mathrm{Fib}}}$ with values in ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ such that ${\ensuremath{\mathrm{Sing}\left({c_{-{\phi}^{2},{\phi}}}\right)}} \cap (-{\phi}^{2},{\phi})=\emptyset$. Then the restriction of $c_{-{\phi}^{2},{\phi}}$ to $(-{\phi}^{2},{\phi})$ is in ${\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^{2},{\phi})_{\omega}$.

Cocycles on ${\ensuremath{\mathrm{Fib}}}$ satisfy the relations in .  Hence $$\begin{aligned}
\label{4tev}
c_{-{\phi}^{2},{\phi}}|(1-T^{-1}ST^{-2}) &\;=\;
c_{-{\phi}^{2},{\phi}}-c_{{\phi}^2,{\phi}} \;=\; c_{-{\phi}^2,{\phi}^2} \\
\nonumber
& \;=\; c_{-{\phi}^{2},-{\phi}} + c_{-{\phi},{\phi}^2} \;=\;
c_{-{\phi}^2,{\phi}}|(T^{-1}-ST^2)\,.\end{aligned}$$ The fact that the restriction $g$ of $c_{-{\phi}^2,{\phi}}$ to $(-{\phi}^2,{\phi})$ has no singularities in $(-{\phi}^2,{\phi})$ implies that relation holds on $(-{\phi},{\phi})$ as an identity of real analytic functions.  Hence $g\in {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}$.

\[prop-thh\]Each element $g\in {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}$ is the restriction $$c_{-{\phi}^2,{\phi}}|_{(-{\phi}^2,{\phi})}$$ for a unique ${\ensuremath{\Gamma}}$-cocycle $c$ on ${\ensuremath{\mathrm{Fib}}}$ with values in ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$. This cocycle satisfies $${\ensuremath{\mathrm{Sing}\left({ c_{-{\phi}^2,{\phi}} }\right)}} \subset \{-{\phi}^2,{\phi}\}\,,\;\;\;
{\ensuremath{\mathrm{Sing}\left({c_{-{\phi},{\phi}^{-1}}}\right)}}\subset \{-{\phi},{\phi}^{-1}\}\,,\;\;\;
{\ensuremath{\mathrm{Sing}\left({c_{-{\phi},{\phi}} }\right)}}\subset\{-{\phi},{\phi}\}\,.$$ The induced map ${\vartheta}:{\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}) \longrightarrow
H^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ is injective. 

First we assume that a cocycle $c$ exists such that $g$ is equal to the restriction of $c_{-{\phi}^2,{\phi}}$ to $(-{\phi}^2,{\phi})$. The cocycle relations imply $$\begin{aligned}
2 g|ST^2 &\;=\; c_{-{\phi},-{\phi}^2}&\quad\text{ on
}&(-{\phi},-{\phi}^2)_c\,,
\displaybreak[0]\\
g|(1+ST^2)&\;=\; c_{-{\phi}^2,{\phi}}+c_{-{\phi},-{\phi}^2}\;=\;
c_{-{\phi},{\phi}}&\text{ on }&(-{\phi},{\phi})\,,
\displaybreak[0]\\
g|(1+ST^2)STS&\;=\; c_{{\phi}^2,{\phi}^{-1}}&\text{ on
}&({\phi}^2,{\phi}^{-1})_c\,,
\displaybreak[0]\\
g|T^{-1}&\;=\; c_{-{\phi},{\phi}^2}&\text{ on }&(-{\phi},{\phi}^2)\,,
\displaybreak[0]\\
g|\left( T^{-1}+(1+ST^2)STS\right)&\;=\;
c_{-{\phi},{\phi}^2}+c_{{\phi}^2,{\phi}^{-1}} \;=\; c_{-{\phi},{\phi}^{-1}}&\text{ on
}&(-{\phi},{\phi}^{-1})\,.\end{aligned}$$ This shows that the restriction of $c_{-{\phi},{\phi}^{-1}}$ to $(-{\phi},{\phi}^{-1})$ is determined by $g$. From we know that $c_{-{\phi},{\phi}^{-1}}|S = - c_{-{\phi},{\phi}^{-1}}$. Hence $g$ determines the restriction of $c_{-{\phi},{\phi}^{-1}}$ to $({\phi}^{-1},-{\phi})_c$ as well.  So $g$ determines $c_{-{\phi},{\phi}^{-1}}$ as an element of ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$.

The situation for the other generator $c_{-{\phi},{\phi}}$ is slightly more complicated.  $$\begin{aligned}
2 c_{-{\phi},{\phi}} &\;=\; g|(1+ST^2) &\quad\text{ on
}&(-{\phi},{\phi})\,,\displaybreak[0]\\
c_{{\phi},{\phi}^{-2}}&\;=\; c_{-{\phi},{\phi}}|T^{-1}S \;=\;
g|(1+ST^2)|T^{-1}S&\quad\text{ on }&
({\phi},{\phi}^{-2})_c\,,\displaybreak[0]\\
c_{{\phi}^{-2},-{\phi}}&\;=\; c_{-{\phi},{\phi}}|ST\;=\; g|(1+ST^2)|ST&\quad\text{
on }&({\phi}^{-2},-{\phi})_c\,.\end{aligned}$$ The relation $c_{-{\phi},{\phi}}+c_{{\phi},{\phi}^{-2}}+c_{{\phi}^{-2},-{\phi}}=0$ implies that $c_{-{\phi},{\phi}}$ is determined by $g$ on each of the cyclic intervals $(-{\phi},{\phi}^{-2})$, $({\phi}^{-2},{\phi})$ and $({\phi},-{\phi})_c$. So any ${\ensuremath{\mathcal{V}}}^{{{\omega}^\ast}}_s$-valued cocycle $c$ on ${\ensuremath{\mathrm{Fib}}}$ is determined by the restriction of $c_{-{\phi}^2,{\phi}}$ to $(-{\phi}^2,{\phi})$.

This reasoning also shows how to construct $c$ from $g$. We put for a given function $g\in {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}$: $$\begin{aligned}
2 \label{hkdef}
h&\;=\;& g|(1+ST^2)&\in {\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi},{\phi})\,,\\
\nonumber
k&\;=\;& g|T^{-1}+h|STS&\in {\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi},{\phi}^{-1})\,.\end{aligned}$$ By the reasoning given above we should have: $$\begin{aligned}
\label{ccdef}
c_{-{\phi},{\phi}^{-1}} &\;:=\; \begin{cases}
k&\text{ on }(-{\phi},{\phi}^{-1})\,,\\
-k|S&\text{ on }({\phi}^{-1},-{\phi})_c\,,
\end{cases}\\
\nonumber
c_{-{\phi},{\phi}}&\;:=\; \begin{cases}
h&\text{ on }(-{\phi},{\phi})\,,\\
-h|(T^{-1}S+ST)&\text{ on }({\phi},-{\phi})_c\,. 
\end{cases}\end{aligned}$$ To see that this indeed defines a cocycle, we choose the base point $-{\phi}$ and consider not the potential cocycle $c$ on ${\ensuremath{\mathrm{Fib}}}$, but the corresponding cocycle ${\psi}$ on ${\ensuremath{\Gamma}}$: $${\psi}_S \;=\; -c_{-{\phi},{\phi}^{-1}}\,,\qquad {\psi}_{T^{-1}S} \;=\; -
c_{-{\phi},{\phi}}\,.$$ The relations turn out to be satisfied.  So indeed there exists a cocycle $c$ as desired. 

For the singularities, we note first that the expressions above for $c_{-{\phi},{\phi}}$ and $c_{-{\phi},{\phi}^{-1}}$ in terms of $g$ imply that $${\ensuremath{\mathrm{Sing}\left({c_{-{\phi},{\phi}^{-1}}}\right)}} \subset \{-{\phi},{\phi}^{-1}\}\,,\qquad {\ensuremath{\mathrm{Sing}\left({
c_{-{\phi},{\phi}}}\right)}} \subset\{-{\phi},{\phi}\}\,.$$

The cocycle $c$ is determined by $c_{-{\phi},{\phi}^{-1}}$ and $c_{-{\phi},{\phi}}$. We now check that the restriction of $c_{-{\phi}^2,{\phi}}$ to $(-{\phi}^2,{\phi})$ is equal to $g$, as desired, and cannot have singularities in $({\phi},-{\phi}^2)_c$. The cocycle relations imply that $$\label{cr} c_{-{\phi}^2,{\phi}}=c_{-{\phi},{\phi}^{-1}}|T+
c_{-{\phi},{\phi}}+c_{-{\phi},{\phi}}|ST \,.$$ On various intervals we have: $$\renewcommand\arraystretch{1.5}
\begin{array}{|c|c|c|c|c|}\hline
&(-{\phi}^2,-{\phi})&(-{\phi},-{\phi}^{-2})&(-{\phi}^{-2},{\phi})&({\phi},-{\phi}^2)_c\\
\hline
c_{-{\phi},{\phi}^{-1}}|T& k|T&k|T&-k|ST&-k|ST \\
c_{-{\phi},{\phi}} &-h|(T^{-1}S+ST)& h&h&-h|(T^{-1}+ST)\\
c_{-{\phi},{\phi}}|ST& h|ST&-h|(1+T^{-1}S)&h|ST&h|ST \\ \hline
c_{-{\phi}^2,{\phi}}& g & g & g
& -k|ST-h|T^{-1}
\\ \hline
\end{array}$$ On $(-{\phi}^{-2},{\phi})$ we use the four term equation: $$\begin{aligned}
c_{-{\phi}^2,{\phi}}&\;=\; - g|T^{-1}ST + h|(-ST^2 + 1+ST)\\
&\;=\; g | \left( -T^{-1}ST + (1+ST^2)(1+ST-ST^2) \right)\\
&\!\!\stackrel{\eqref{4te|}}=\; - g|(1+ST^2-T^{-1}ST^{-2})ST +
g|\left( 1-ST^2ST^2+ST+ST^2ST\right)\\
&\;=\; g|\left( T^{-1}ST^{-2}ST+1-ST^2ST^2\right)
\;\stackrel{\eqref{B1.1}}=\; g\,.\end{aligned}$$ Hence $c_{-{\phi}^2,{\phi}}= g$ on $(-{\phi}^2,-{\phi}) \cup (-{\phi},-{\phi}^{-2})
\cup( -{\phi}^{-2},{\phi})$. Since $g$ is analytic on $(-{\phi}^2,{\phi})$, the points $-{\phi}$ and $-{\phi}^{-2}$ are not singularities of $c_{-{\phi}^2,{\phi}}$, by the definition of ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ as an inductive limit.  Furthermore $c_{-{\phi}^2,{\phi}}$ is given by the analytic function $-k|ST-h|T^{-1}$ on $({\phi},-{\phi}^2)_c$. This shows that ${\ensuremath{\mathrm{Sing}\left({c_{-{\phi}^2,{\phi}}}\right)}} \subset\{-{\phi}^2,{\phi}\}$.

To show that the map ${\vartheta}: {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}\longrightarrow H^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{\omega}^\ast}}_s)$ given by ${\vartheta}: g \mapsto [c]$ is injective, we check that the cocycle $c$ corresponding to $g$ can be a coboundary only if $g=0$. If $c$ is a coboundary, then $c_{{\gamma}^{-1}(-{\phi}),{\delta}^{-1}(-{\phi})} = v|{\delta}- v|{\gamma}$. In particular, $v|TST^2$ should be equal to $v$ for some $v\in
{\ensuremath{\mathcal{V}}}^{{{\omega}^\ast}}_s$.

We use that $TST^2$ is a hyperbolic element of ${\ensuremath{\Gamma}}$ fixing $-{\phi}$ and ${\phi}^{-1}$. Conjugation in ${\ensuremath{{\mathrm{PSL}\!\left(2, {\ensuremath{\mathbb{R}}}\right)}}}$ transforms it to $\eta={{\begin{bmatrix} {\phi}^2 & 0 \\ 0 & {\phi}^{-2}
\end{bmatrix}}}$, fixing $0$ and $\infty$. Let $w\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ be invariant under $\eta$. The action of $\eta$ on $(0,\infty)$ and $(-\infty,0)$ is by $x\mapsto  {\phi}^4 x$. So if $w$ were to have singularities in ${\ensuremath{\mathbb{R}}}{\smallsetminus}\{0\}$, then there would be infinitely many, contradicting the definition of ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$. On ${\ensuremath{\mathbb{R}}}$ we have $w|\eta(x) = {\phi}^{4s} w({\phi}^4x)$. Inserting this into a power series expansion converging on a neighborhood of $0$, we see that if $w$ is analytic at $0$ it vanishes.  The same holds at $\infty$. So if $w\neq 0$, then ${\ensuremath{\mathrm{Sing}\left({w}\right)}} = \{0,\infty\}$.

Conjugating back, we see that if $c$ is a non-zero coboundary, then it has the form $c_{{\gamma}^{-1}(-{\phi}),{\delta}^{-1}(-{\phi})} = v|({\delta}-\nobreak{\gamma})$, where $v\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ satisfies $v|TST^2=v$ and ${\ensuremath{\mathrm{Sing}\left({v}\right)}} = \{-{\phi},{\phi}^{-1}\}$. We consider the singularities of $c_{-{\phi}^2,{\phi}} = c_{T^{-1}(-{\phi}),TS(-{\phi})} = v|ST^{-1}-v|T$. Now ${\ensuremath{\mathrm{Sing}\left({v|ST^{-1}}\right)}}=\{{\phi},-{\phi}^{-1}\}$, and ${\ensuremath{\mathrm{Sing}\left({v|T}\right)}} = \{-{\phi}^2, -{\phi}^{-2}\}$. So $c_{-{\phi}^2,{\phi}}$ has singularities at $-{\phi}^{-1}$ and $-{\phi}^{-2}$, in contradiction to the analyticity of $g$ on $(-{\phi}^2,{\phi})$.

We note that $H^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ is a subspace of $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$. The inclusion ${\ensuremath{\mathcal{V}}}^{\omega}_s\hookrightarrow{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ induces a natural map $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s) \longrightarrow  H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$.

\[prop-om\]The subspace ${\vartheta}({\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}) $ of $ H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ is contained in the image of $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s)$ in $ H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$.

Let $g\in {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}$, and let $c$ be the cocycle on ${\ensuremath{\mathrm{Fib}}}$ representing ${\vartheta}({\gamma})$. Since ${\ensuremath{\mathrm{Sing}\left({c_{-{\phi},{\phi}^{-1}}}\right)}} \subset \{ -{\phi},{\phi}^{-1}\}$, Proposition \[D2.6\] implies that there are $K_{-{\phi}}$ and $K_{{\phi}^{-1}}$ in ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}$ with ${\ensuremath{\mathrm{Sing}\left({K_{-{\phi}}}\right)}}\subset\{-{\phi}\}$ and ${\ensuremath{\mathrm{Sing}\left({K_{{\phi}^{-1}}}\right)}}\subset\{{\phi}^{-1}\}$ such that $c_{-{\phi},{\phi}^{-1}} = K_{-{\phi}}-K_{{\phi}^{-1}}$. Since $c_{-{\phi},{\phi}^{-1}}|(1+\nobreak S)=0$, we have $K_{{\phi}^{-1}} = K_{-{\phi}}|S + {\alpha}$ for some ${\alpha}\in {\ensuremath{\mathcal{V}}}^{\omega}_s$. Similarly, there are $H_{-{\phi}}, H_{\phi}\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$ with ${\ensuremath{\mathrm{Sing}\left({H_{-{\phi}}}\right)}}\subset\{-{\phi}\}$ and ${\ensuremath{\mathrm{Sing}\left({H_{\phi}}\right)}}\subset \{{\phi}\}$ such that $c_{-{\phi},{\phi}} = H_{-{\phi}}-H_{\phi}$. There exists a function ${\beta}\in {\ensuremath{\mathcal{V}}}^{\omega}_s$ such that $H_{\phi}= H_{-{\phi}}|T^{-1}S+{\beta}$.

We have, with : $$c_{-{\phi}^2,{\phi}} \;\in\;c_{-{\phi},{\phi}^{-1}}|T + c_{-{\phi},{\phi}}|(1+\nobreak
ST)
\in K_{-{\phi}}|(T-\nobreak ST) + H_{-{\phi}}|(ST-\nobreak T^{-1}S) +
{\ensuremath{\mathcal{V}}}^{\omega}_s \,.$$ Since ${\ensuremath{\mathrm{Sing}\left({c_{-{\phi}^2,{\phi}}}\right)}}\subset\{-{\phi}^2,{\phi}\}$, we conclude from the singularities of the various terms that $K_{-{\phi}}|ST \in H_{-{\phi}}|ST +{\ensuremath{\mathcal{V}}}^{\omega}_s$. Hence $$K_{-{\phi}}-H_{-{\phi}}\in {\ensuremath{\mathcal{V}}}^{\omega}_s\,.$$

The class $[c]\in H^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ considered as a class in $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ is given by the group cocycle $${\psi}_S \;=\; -c_{-{\phi},{\phi}^{-1}}\,,\qquad {\psi}_{T^{-1}S} \;=\; -
c_{-{\phi},{\phi}}\,.$$ We add to it the coboundary ${\gamma}\mapsto K_{-{\phi}}|(1-\nobreak{\gamma})$, obtaining a cocycle $\tilde{\psi}$ in the same class.  It satisfies $$\begin{aligned}
\label{tpsi-on-gen}
\tilde{\psi}_S &\;=\;
-K_{-{\phi}}+K_{{\phi}^{-1}}+K_{-{\phi}}|(1-S) \;=\; {\alpha}\;\in\; {\ensuremath{\mathcal{V}}}^{\omega}_s\,,\\
\nonumber
\tilde{\psi}_{T^{-1}S} &\;=\;
-H_{-{\phi}}+H_{\phi}+ K_{-{\phi}}|(1-\nobreak T^{-1}S) \\
\nonumber
&\;=\; (K_{-{\phi}}-\nobreak H_{-{\phi}})|(1-\nobreak T^{-1}S) +{\beta}\;\in\;
{\ensuremath{\mathcal{V}}}^{\omega}_s\,.\end{aligned}$$ Thus the class $[{\psi}] = [\tilde{\psi}]$ in $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{\omega}^\ast}}_s)$ is the image of the class $[\tilde{\psi}]\in H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s)$.

\[prop-thsurj\]The subspace ${\vartheta}\left( {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}\right)$ of $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ is equal to the image of $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s)$.

We use a hyperbolic one-sided average, as discussed at the end of §\[C1\].  Starting with a cocycle $\tilde {\chi}\in Z^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s)$ we define $$A= \tilde {\chi}_{TST^2} | {\mathrm{A\!v}}_{TST^2}^+ \in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s\,,$$ with ${\ensuremath{\mathrm{Sing}\left({A}\right)}} \subset \{-{\phi}\}$ and $A|(1-\nobreak TST^2) = \tilde {\chi}_{TST^2}$. We have used that $-{\phi}$ is the repelling fixed point of $TST^2$.

Now ${\chi}:{\gamma}\mapsto \tilde{\chi}_{\gamma}- A|(1-\nobreak{\gamma})$ defines a ${\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$-valued cocycle in the same cohomology class in $H^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ as $\tilde {\chi}$. It satisfies ${\ensuremath{\mathrm{Sing}\left({{\chi}_{\gamma}}\right)}}\subset \{-{\phi},{\gamma}^{-1}(-{\phi})\}$. Moreover, ${\chi}_{TST^2}=0$, so ${\chi}$ is hyperbolic for the conjugacy class of $TST^2$. Hence it corresponds to a cocycle $c$ on ${\ensuremath{\mathrm{Fib}}}$, such that $c_{-{\phi}^2,{\phi}} = A|T - A|ST^{-1}-\tilde{\chi}_{ST^{-2}}|T$ has singularities at most in $\{-{\phi}^2,{\phi}\}$. This means that the class of $\tilde {\chi}$ is in the image of ${\vartheta}$.

Starting from this cocycle $c$, one can take $K_{-{\phi}}=H_{-{\phi}} = A$, and get back $\tilde{\psi}= \tilde{\chi}$.

The results in this subsection imply Theorem \[thm-th\]. 

Eigenfunctions of the transfer operator and cohomology {#sect-eif-coh}
------------------------------------------------------

In this final section we prove Theorem \[thm-par\]. 

We use the same notations as in the previous subsection: $g\in
{\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}$, $c\in Z^1_{\ensuremath{\mathrm{Fib}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$, $K_{-{\phi}}\in {\ensuremath{\mathcal{V}}}^{{{\omega}^\ast}}_s$ with ${\ensuremath{\mathrm{Sing}\left({K_{-{\phi}}}\right)}} \subset\{-{\phi}\}$, ${\psi}\in Z^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s)$ and $\tilde{\psi}\in Z^1({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s)$, related by: $$\begin{aligned}
\label{g-c}
g&\;=\; c_{-{\phi}^2,{\phi}}\text{ restricted to }(-{\phi}^2,{\phi})\,,\\
\label{c-ps}
c_{{\gamma}^{-1}(-{\phi}),{\delta}^{-1}(-{\phi})} &\;=\; {\psi}_{{\gamma}{\delta}^{-1}}|{\delta},\qquad
{\psi}_{\gamma}\;=\;
-c_{-{\phi},{\gamma}^{-1}(-{\phi})}\,,\\
\label{ps-tps}
\tilde{\psi}_{\gamma}&\;=\; {\psi}_{\gamma}+ K_{-{\phi}}|(1-{\gamma})\,,\\
\label{K-ps}
K_{-{\phi}}&\;=\; \tilde{\psi}_{TST^2}|{\mathrm{A\!v}}_{TST^2}^+\,.\end{aligned}$$

We consider $P\in {\ensuremath{\mathcal{V}}}^{\omega}_s(-{\phi}^2,{\phi})$ given by $$P\;=\; g|ST^3{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}-g|T^{-1}ST^{-1}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}-g\,.$$ So $P$ measures how much $(g,g|T^{-1})$ differs from an $1$-eigenfunction of ${\ensuremath{{\ensuremath{\mathcal{L}}}}}$ in .  Application of shows that $P|T=P$ on $(-{\phi}^2,{\phi}^{-1})$. Hence the periodic function $P$ extends as an element of ${\ensuremath{\mathcal{V}}}^{\omega}_s({\ensuremath{\mathbb{R}}})^T \subset{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast}}}_s$. To obtain a representation of $P$ on ${\ensuremath{\mathbb{R}}}$, we use – to obtain on $(-{\phi}^2,{\phi})$: $$\begin{aligned}
P&\;=\; {\psi}_{TST}|T^{-1}S \left( ST^3{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}-T^{-1}ST^{-1}{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}-1\right)\\
&\;=\; \tilde{\psi}_{TST}|\left( T^2{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}-S{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}-T^{-1}S\right)\\
&\qquad\hbox{} \nobreak
+ K_{-{\phi}}|\left( \left(TST^3 - T^2\right){\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}+ \left(
S-ST^{-1}\right)
{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}+ T^{-1}S-T\right)\,.\end{aligned}$$ From it follows that $K_{-{\phi}} |TST^2 = K_{-{\phi}} - \tilde{\psi}_{TST^2}$. With we find for the contribution of $K_{-{\phi}}$: $$\begin{aligned}
K_{-{\phi}}|&(T-T^2){\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}- \tilde{\psi}_{TST^2}|T{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}-K_{-{\phi}}|ST^{-1}(1-T){\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}+ K_{-{\phi}}|(T^{-1} S-T)\\
&\;=\; K_{-{\phi}}|\left( T - ST^{-1} +T^{-1}S-T\right)
-\tilde{\psi}_{TST^2}|T{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}\\
&\;=\; \left( K_{-{\phi}}|TST^2 + \tilde{\psi}_{TST^2}\right)|T^{-1}S -
K_{-{\phi}}|ST^{-1} - \tilde{\psi}_{TST^2}|T{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}\\
&\;=\; \tilde{\psi}_{TST^2}|(T^{-1}S-T{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}})\,.\end{aligned}$$ This leads to an expression valid on ${\ensuremath{\mathbb{R}}}$: $$\begin{aligned}
\nonumber
P&\;=\; \tilde{\psi}_{TST}|\left( T^2 {\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}-S{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}-T^{-1}S +
T(T^{-1}S-T{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}})\right)
\\
\nonumber
&\qquad\hbox{}
+ \tilde{\psi}_T| (T^{-1}S +1 -{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}})\\
\label{P-glob}
&\;=\; \tilde{\psi}_{TST}|(S-T^{-1}S) + \tilde{\psi}_T|(T^{-1}S+1)
- \tilde{\psi}_T |{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}-\tilde{\psi}_{TST}|S{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\,.\end{aligned}$$

Suppose that $P=0$, i.e., $g\in {\ensuremath{{}^4\mathrm{FE}}}_s(-{\phi}^2,{\phi})_{\omega}$ corresponds to an eigenfunction $(g,g|T^{-1})$ of ${\ensuremath{{\ensuremath{\mathcal{L}}}}}$. We note that $\tilde {\psi}_{TST}, \tilde{\psi}_T \in {\ensuremath{\mathcal{V}}}^{\omega}_s$, and apply Lemma \[C1.8\] to conclude that $\tilde{\psi}_T
|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}= \tilde{\psi}_T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}\in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s$, hence $\tilde {\psi}_T \in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s|(1-\nobreak T)$, which is the condition for a cocycle to be parabolic.  Thus we conclude that $[\tilde{\psi}] \in H^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$.

Conversely, suppose that $\tilde{\psi}\in Z^1_{{\mathrm{par}}}({\ensuremath{\Gamma}};{\ensuremath{\mathcal{V}}}^{\omega}_s,{\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s)$. By parabolicity $\tilde{\psi}_T \in {\ensuremath{\mathcal{V}}}^{{{{\omega}^\ast\!,\mathrm{simple}}}}_s|(1-\nobreak T)$, and hence $\tilde{\psi}_T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^+}}=\tilde{\psi}_T|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$ by Lemma \[8.17\].  In we see that $P$ is in ${\ensuremath{\mathcal{V}}}^{\omega}_s + {\ensuremath{\mathcal{V}}}^{\omega}_s|{\ensuremath{{\ensuremath{{\mathrm{A\!v}}_T}}^-}}$. Hence $P(x)$ has an asymptotic behavior as $x\downarrow-\infty$ of the form indicated in .  Since $P$ is also periodic, it vanishes. 

[99]{}

E. Artin, *Ein mechanisches System mit quasi-ergodischen Bahnen*, Abh.  Math.  Sem.  d.Hamburgischen Universität, 3 (1924), 170–175

R.W. Bruggeman, J. Lewis and D. Zagier, *Period functions for [M]{}aass wave forms. [ II]{}: Cohomology*, in preparation. 

L. Hörmander, *An introduction to complex analysis in several variables*, D. Van Nostrand Co., Inc., Princeton, N.J., USA, 1966. 

A. Hurwitz, *Über eine besondere Art der Kettenbruch-Entwicklung reeller Grössen*, Acta Mathematica **12** (1889), 367–405. [ `DOI:10.1007/BF02391885`](http://dx.doi.org/10.1007/BF02391885)

J. Lewis and D. Zagier, *Period functions for [M]{}aass wave forms. [ I]{}*, Annals of Mathematics **153** (2001), 191–258.  Euclid Identifier: [`euclid.annm/1026916788`](http://projecteuclid.org/getRecord?id=euclid.annm/1026916788)

D. Mayer, *Transfer operators, the Selberg-zeta function and Lewis-Zagier theory of period functions*, Lecture notes of a course given in Günzburg. [ `http://www.dynamik.tu-clausthal.de/research/preprints/Mayer.001.ps`](http://www.dynamik.tu-clausthal.de/research/preprints/Mayer.001.ps)

D. Mayer, *Continued fractions and related transformations*, In Tim Bredford, Michael Keane, and Caroline Series, editors, Ergodic theory, Symbolic Dynamics abd Hyperbolic Spaces, pages 175–222.  Oxford University Press, 1991. 

[^1]: The second named author was supported in by the [Deutsche Forschungsgemeinschaft](http://www.dfg.de/) through the DFG Research Project “Transfer operators and non arithmetic quantum chaos” (Ma 633/16-1).