数理解析研究所講究録 1220 巻 2001 年 141-143

141

ON THE CONSTRUCTION OF CONFORMAL MEASURESFOR PIECEWISE-INVERTIBLE SYSTEMS$C^{0}$MICHIKO YURIABSTRACT. We present anew method for the construction of conformal measures
for infinite to one piecewice-invertible Markov systems. We direct our attention
to potentials which may fail both summable variations and bounded distortion but
satisfy the weak bounded variation. Our results apply to higher-dimensional maps
which are not necessarily conformal and admit certain nonhyperbolic periodic orbits.

$\nu$$C^{0}$$\phi$\S 0 Introduction
Let $(T, X, Q=\{X_{i}\}_{i\in I})$ be apiecewise
Convertible system i.e., $X$ is acompact
metric space with metric ,: $Xarrow X$ is anoninvertible map which is notnecessarily continuous, and $Q=\{X_{i}\}_{i\in I}$ is acountable disjoint partition $Q=$
of $X$ such thatis dense in $X$ and satisfy the following properties.
(01) For each $i\in I$ with,: $intX_{i}arrow T(intX_{i})$ is ahomeomorphism andextends to ahomeomorphism on $d(T(intX_{i}))$ .
(02).
(03)
generates , the sigma algebra of Borel subsets of $X$ .
Letsatisfy $int(X_{i_{1}}\cap T^{-1}X_{i_{2}}\cap\ldots T^{-(n-1)}X_{i_{n}})\neq\emptyset$. Thenwe define
which is called acylinder of rank
and write. By (01),:ishomeomorphism andextends to ahomeomorphism:.
We impose on $(T, X, Q)$ the next condition which gives anice countable states
symbolic dynamics similar to sofic shifts (cf. [5]):$C^{0}$$d$$\{X_{i}\}_{i\in I}$$T$$\bigcup_{i\in I}intX_{i}$$intX_{i}\neq\emptyset$$T|_{intX_{i}}$$(T|_{intx_{:}})^{-1}$$v_{i}$$T( \bigcup_{intx_{:=\emptyset}}X_{i})\subset\bigcup_{intx_{:=\emptyset}}X_{i}$$\{X_{i}\}_{i\in I}$ ‘$\underline{i}=$$\mathcal{F}$$(i_{1}\ldots i_{n})\in I^{n}$$X_{\underline{i}}:=X_{i_{1}}\cap T^{-1}X_{i_{2}}\cap\ldots T^{-(n-1)}X_{i_{n}}$$n$$|\underline{i}|=n$$T^{n}|_{int\mathrm{x}_{:_{1}..:_{n}}}$$intX_{i_{1}\ldots i_{n}}arrow T^{n}(int(X_{i_{1}\ldots i_{n}}))$$(T^{n}|_{intx_{:_{1}:_{n}}}\ldots)^{-1}$$v_{i_{n}}=v_{i_{1}\ldots i_{n}}$$\mathrm{a}$$v_{i_{1}}\circ v_{i_{2}}\circ\ldots\circ$$cl(T^{n}(intX_{\underline{i}}))arrow cl(intX_{\underline{i}})$(Finite Range Structure).
finitely many open subsets$\mathcal{U}=\{int(T^{n}X_{i_{1}\ldots i_{n}}) :$U_{1}\ldots$ $U_{N}$of$X$.

\forall X_{i_{1}\ldots i_{n}},\forall n>0\}$consists ofIn particular, we say that $(T, X, Q)$ satisfies Bernoulli property if $cl(T(intX_{i}))=$$X(\forall i\in I)$ so that
{intX} and that $(T, X, Q)$ satisfies Markov property if$(intX_{i})\cap
cl$int(cl(intTXj))\neq \emptyset implies $cl(intTX_{j})\supset cl(intX_{i})$ . $(T, X, Q)$ satisfyingBernoulli (Markov) property is called apiecewise-invertible Bernoulli (Markov)
system respectively.
$\mathcal{U}=$$C^{0}$

1991 Mathematics Subject Classification.
.

$28\mathrm{D}99,28\mathrm{D}20,58\mathrm{F}11,58\mathrm{F}03,37\mathrm{A}40,37\mathrm{A}30,37\mathrm{C}30$,$37\mathrm{D}35,37\mathrm{F}10,37\mathrm{A}45$

Typeset by$\mathrm{A}\Lambda 4\theta \mathrm{I}\mathrm{E}\mathrm{K}$142’-M. YURI

be the first return function over:For given asubset of $X$ , letand we define $D_{n}^{A}=\{x\in A : R_{A}(x)>n\}$ . If we have previously areasonable
algebra of Borel, where denotes themeasurable dynamics e.g.
$(\nuT^{-1}\sim\nu)$probability measure with $\nu(A)>0$
subsets of $X$ and is anonsingularsatisfying(1) :’$A$$R_{A}$$Aarrow \mathrm{N}\cup\infty$$A$$(T, X, Q, \mathcal{F}, \nu)$$\mathcal{F}$$\sigma-$$\nu$$\lim_{narrow\infty}\nu(D_{n}^{A})=\nu(\cap D_{n}^{A})=0n\geq 0$

andover can be defined almost everywhere onthen the induced map-invariant er, too. Furthermore, if we can construct aiterations
absolutely continuous with respect to , then thegodic probability measureintegrability of the first return function with respect to (which is equivalent toso that (1) is automatically satisfied) is sufficient for the(2) :existence of -invariant ergodic probability measure absolutely continuous withwhich is given by the well-known Kac formula as follows :for all
respect toau

$A$

$A$

$T_{A}$$T_{A}$$\{T_{A}^{n}\}_{n\geq 1}$$\nu$$\mu A$$\nu$$\sum_{n\geq 0}\nu(D_{n}^{A})<\infty$$T$$\mu$$\nu$$f \in L^{1}(\nu)\frac{1}{\mu(A),\mathrm{s}\mathrm{u}},\int_{1\mathrm{T}\mathrm{h}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{t}\mathrm{s}}Xfd\mu=\int Af_{A}d\mu_{A,\mathrm{d}},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}f_{A}(X)=\sum_{\mathrm{f}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}1\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{y}\mathrm{n}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}1\mathrm{i}\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}1}\dot{\mathrm{t}}=^{fT^{\dot{l}}(x)}0R_{A}(x)-1$

ofweak bounded variation were established in [6] for piecewise -invertible Bernoulli
and on the associated potensystems by assuming some regulalr condition on
satisfactory
to establish Thermodynamic
tial . In particular, this approach works
formalism for piecewise -invertible maps with the Bernoulli property admitting
periodic points) and for thecertain nonhyperbolic periodic orbits (ural potential $=-\log|detDT|$ . In fact, can be taken as ahyperbolic region
which is away from the nonhyperbolic periodic orbits (see [6] for details) and theabsolutely continuous invariant measure with respect to the normalized Lebesgue
measure attains the measure theoretical pressure for $=-\log|detDT|$ . On theother hand, when $(T, X, Q)$ does not satisfy the Bernoulli property we have noevidence of the existence of nonsingular reference measure even if the Markov
property is satisfied. If we restrict our attention to (countable) Markov shifts thenwe can find some answer to this problem (e.g.,[4]). However, if the systems are notsymbolic dynamics the existence problem is still remain open (cf.[l]). In this talk,for infinite to one piecewise -invertible transitive Markov systems we shall giveapartial answer to this problem. For this purpose, we first clarify properties ofdefined on asingle
topological pressure for and for the associated potential
overcylinder $A\in Q$ . Then we introduce Schweiger’s jump transformations$X$
betweencylinders which are mapped onto under . We shall see agood relation
associated toand the topological pressure forthe topological pressure for. This observation allows one to establish the existence of an eigenvalue 1of theby using aformula of zeta function
Perron-Frobenius operator associated toobtained in [5]. FinaUy we can construct afor in terms of zeta function forconformal measure by using aresult in [1]. We also establish the existence of conformal measures by using jump transformation defined. Again the existence of anplays an importanteigenvalue 1of the Perron-Frobenius operator associated torole for the construction of .
$\phi$$C^{0}$$T_{A}$$\phi_{A}$$C^{1}$$\mathrm{e}.\mathrm{g}.,\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}$$A$$\mathrm{n}\mathrm{a}\acute{\mathrm{t}}$$\phi$$\mu$$\phi$

$\nu$

$\nu$

$C^{0}$$\phi_{A}$$\phi$$T^{*}$$T$$\phi^{*}$$\phi_{A}$$T^{*}$$\phi_{A}$$\phi^{*}$$\phi$$\nu$$\phi^{*}$$\nu$

REFERENCES
1. M. Denker and M. Yuri, A note on the construction of nonsingular Gibbs measures, Coll-143
EQUILIBRIUM STATES FOR PIECEWISE INVERTIBLE SYSTEMS
quium Mathematicum 84/85 (2000), 377-383.
2. P.H.Harms, R.D. Mauldin and M. Urbanski, Thermodynamic Formalism andmulti-fractal
analysis of conformal infinite iterated functional systems, Preprint.
3. R.D. Mauldin and M. Urbanski, Parabolic iterated functional systems, Preprint.
4. Omri Sarig, Thermodynamic Formalism for countable Markov shifts., Ergodic Theory andDyn. Syst. 19 (1999), 1565-1593.
5. M. Yuri, Zeta functions for certain nonhyperbolic systems and topological Markov approxirnations, Ergodic Theory and Dyn. Syst. 18 (1998), 1589-1612.
6. M. Yuri, Thermodynamic formalism for certain nonhyperbolic maps, Ergodic Theory andDyn. Syst. 19 (1999), 1365-1378.
7.
Weak Gibbs measures for certain nonhyperbolic systems, Ergodic Theory and Dyn.
Syst. 20 (2000), 1495-1518.
8. M. Yuri, Equilibrium states for piecewise invertible systems associated to potentials of weakbounded variation. , Preprint.

YURI:
DEPARTMENT OF BUSINESS ADMINISTRATION, Sapporo UNIVERSITY, NISHIOKA,TOYOHIRA-KU,
062, JAPAN.
E-mail address: yuri@math.sci.hokudai.ac.jp, yuri@mail-ext.sapp0r0-u.ac.jp$\mathrm{s}_{\mathrm{A}\mathrm{P}\mathrm{P}\mathrm{O}\mathrm{R}\mathrm{O}}$