bluuebunny/crossref_metadata_2025 huggingface dataset:
Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques
Authors: Lesigne Emmanuel
Published: September 2008

We call nilmanifold every compact space X on which a connected locally compact nilpotent group acts transitively. We show that, if X is a nilmanifold and f is a continuous function on X, then, for all x in X and a in N, the sequence

converges. We give a process for the computation of the limit. A similar result for the continuous means is presented.
The C1 closing lemma for non-singular endomorphisms
Authors: Wen Lan
Published: September 2008

A slightly improved version of the (idealized) C1 closing lemma is proved. This turns out to generalize the C1 closing lemma from diffeomorphisms to nonsingular endomorphisms.
The action of inert finite-order automorphisms on finite subsystems of the shift
Authors: Boyle Mike , Fiebig Ulf-Rainer
Published: September 2008

Let (X, S) be a shift of finite type. Let G be the group of automorphisms of (X, S) which are compositions of elements of finite order in the kernel of the dimension representation. We characterize the action of G on finite subsystems of (X, S).
Rigidity of symplectic Anosov diffeomorphisms on low dimensional tori
Authors: Flaminio L. , Katok A.
Published: September 2008

We show that any symplectic Anosov diffeomorphism of a four torus T4 with sufficiently smooth stable and unstable foliations is smoothly conjugate to a linear hyperbolic automorphism of T4.
Exceptional minimal sets of C1+α-group actions on the circle
Authors: Hurder S.
Published: September 2008

We prove two extensions of Sacksteder's Theorem for the action A: Γ × S1 → S1 of a finitely-generated group Γ on the circle by C1+α-diffeomorphisms. If the action A has an exceptional minimal set K with a gap endpoint of exponential orbit growth rate, or if the action A on K has positive topological entropy, then the exceptional set K is hyperbolic. That is, A has a linearly contracting fixed-point in K. A key point of the paper is to prove a foliation closing lemma using the foliation geodesic flow technique.
Poisson limit law for Markov chains
Authors: Pitskel B.
Published: September 2008

For a mixing stationary Markov chain we prove a Poisson limit law for the recurrence to small cylindrical sets. Since hyperbolic torus automorphisms are Markov chains, the result carries over to these transformations.
Smooth Markov partitions and toral automorphisms
Authors: Cawley Elise
Published: September 2008

We show that the only hyperbolic toral automorphisms f for which there exist Markov partitions with piecewise smooth boundary are those for which a power fk is linearly covered by a direct product of automorphisms of the 2-torus. Only a finite number of shapes occur in a certain natural set of cross-sections of the partition boundary. The behavior of the stratified structure of a piecewise smooth boundary under the mapping forces these shapes to be self-similar. This, together with expanding properties of the mapping, means that a piecewise smooth partition is in fact piecewise linear. Orbits of affine disks in the boundary are used to construct a basis of 2-dimensional invariant toral subgroups, and then the product decomposition of a covering follows easily.
Random iterations of rational functions
Authors: Fornaess John Erik , Sibony Nessim
Published: September 2008

We study the asymptotic behavior of iterates of rational functions with small perturbations. In presence of attractive cycles we show that almost surely, in the parameter space, the iterates converge to a given neighborhood of the attractive cycles. When there is no attractive cycle, we prove an ergodic theorem with respect to Lebesgue measure.
Groups acting freely on R-trees
Authors: Morgan John W. , Skora Richard K.
Published: September 2008

In this paper we study the question of which groups act freely on R-trees. The paper has two parts. The first part concerns groups which contain a non-cyclic, abelian subgroup. The following is the main result in this case.

Let the finitely presented group G act freely on an R-tree. If A is a non-cyclic, abelian subgroup of G, then A is contained in an abelian subgroup A′ which is a free factor of G.

The second part of the paper concerns groups whch split as an HNN-extension along an infinite cyclic group. Here is one formulation of our main result in that case.

Let the finitely presented group G act freely on an R-tree. If G has an HNN-decomposition

where (s) is infinite cyclic, then there is a subgroup H′ ⊂ H such that either

(a); or

(b),where S is a closed surface of non-positive Euler characteristic.

A slightly different, more precise result is also given.
Entropy of dynamical systems and perturbations of operators
Authors: Voiculescu Dan
Published: September 2008

In the papers [9, 10, 3, 11] on perturbations of Hilbert space operators, we studied an invariant (τ) where is a normed ideal of compact operators and τ a family of operators. The size of an ideal for which (τ) vanishes or does not vanish is an upper, respectively lower, bound for a kind of dimension of τ. In the case of systems of commuting self-adjoint operators τ, the results of [9,3] relate (τ) with (an ideal slightly smaller than the Schatten von Neumann class ) to the way the spectral measure of τ compares to p-dimensional Hausdorff measure.
Integral formulas for the Laplacian along the unstable foliation and applications to rigidity problems for manifolds of negative curvature
Authors: Yue Chengbo
Published: September 2008

We obtain a class of integral formulas for the Lapacian along unstable leaves of the geodesic flow of compact manifolds of negative curvature. Using these formulas, we give two functional descriptions of those manifolds with horospheres having constant mean curvature. More rigidity problems are discussed, including a simple proof of two important Lemmas by Hamenstadt which avoids her use of stochastic process.
The return times and the Wiener—Wintner property for mean-bounded positive operators in Lp
Authors: Assani I.
Published: September 2008

We prove the following two results for mean-bounded positive operators on Lp(µ) (1
∞).

(1) If (X, , µ, ϕ) is a dynamical system and f ∈ L∞ (X) then the sequence f(ϕn x) is a.e. a universal good sequence for mean-bounded positive operators in Lp. (Return times property.)

(2) If T is a mean-bounded positive operator on LP(X, , µ) and f ∈ Lp (µ) then the sequence Tnf)(x) is a.e. a universal good sequence for all dynamical systems (Y, , v,S) in L∞(v). A corollary of (2) is a Wiener-Wintner property for mean-bounded positive operators on Lp.
Geometric measures for parabolic rational maps
Authors: Denker M. , Urbański M.
Published: September 2008

Let h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.
Time-preserving conjugacies of geodesic flows
Authors: Hamenstädt Ursula
Published: September 2008

In this note we study Borel-probability measures on the unit tangent bundle ofa compact negatively curved manifold M that are invariant under the geodesic flow. We interpret the entropy of such a measure as a Hausdorff dimension with respect to a natural family of distances on the ideal boundary of the universal covering of M. This in term yields necessary and sufficient conditions for the existence of time preserving conjugacies of geodesic flows.
There are no minimal homeomorphisms of the multipunctured plane
Authors: Handel Michael
Published: September 2008

The main results of this paper are the following theorem and its corollary.

Theorem 0.1. Suppose that f: S2 → S2 is an orientation-preserving homeomorphism of the two-dimensional sphere and that Fix (f) is a finite set containing at least three points. If f has a dense orbit then the number of periodic points of period n for some iterate of f grows exponentially in n.
Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties
Authors: Pesin Ya. B.
Published: September 2008

We introduce a class of dynamical systems on a Riemannian manifold with singularities having attractors with strong hyperbolic behavior of trajectories. This class includes a number of famous examples such as the Lorenz type attractor, the Lozi attractor and some others which have been of great interest in recent years. We prove the existence of a special invariant measure which is an analog of the Bowen-Ruelle-Sinai measure for classical hyperbolic attractors and study the ergodic properties of the system with respect to this measure. We also describe some topological properties of the system on the attractor. Our results can be considered a dissipative version of the theory of systems with singularities preserving the smooth measure.
Lyapunov maps, simplicial complexes and the Stone functor
Authors: Robbin Joel W. , Salamon Dietmar A.
Published: September 2008

Let be an attractor network for a dynamical system ft: M → M, indexed by the lower sets of a partially ordered set P. Our main theorem asserts the existence of a Lyapunov map ψ:M → K(P) which defines the attractor network. This result is used to prove the existence of connection matrices for discrete-time dynamical systems.
A new proof of the Brouwer plane translation theorem
Authors: Franks John
Published: September 2008

Let f be an orientation-preserving homeomorphism of ℝ2 which is fixed point free. The Brouwer ‘plane translation theorem’ asserts that every x0 ∈ ℝ2 is contained in a domain of translation for f i.e. an open connected subset of ℝ2 whose boundary is L ∪ f(L) where L is the image of a proper embedding of ℝ in ℝ2, such that L separates f(L) and f−1(L). In addition to a short new proof of this result we show that there exists a smooth Morse function g: ℝ2 → ℝ such that g(f(x)) < g(x) for all x and the level set of g containing x0 is connected and non-compact (and hence the image of a properly embedded line).
When is an Anosov flow geodesic?
Authors: Green Leon W.
Published: September 2008

Let X, H+, H− be vector fields tangent, respectively, to an Anosov flow and its expanding and contracting foliations in a compact three-dimensional manifold, with γ, α+, α− one forms dual to them. If α+([H+, H−]) = α−([H+, H−]) and γ([H+, H−]) = α−([X, H−]) − α+([X, H+]), then the manifold has the structure of the unit tangent bundle of a Riemannian orbifold with geodesic flow field X.
Diffeomorphisms in ℱ1(M) satisfy Axiom A
Authors: Hayashi Shuhei
Published: September 2008

R. Mañé has given a proof of the C1 Stability Conjecture and conjectured that every element of ℱ1(M) satisfies Axiom A. Here we prove that this conjecture is true.
Transfer operators for coupled map lattices
Authors: Keller Gerhard , Künzle Martin
Published: September 2008

Let L denote a finite or infinite one-dimensional lattice. To each lattice site is attached a copy of a dynamical system with phase space [0, 1] and dynamics described by a transformation τ: [0, 1] → [0, 1], which is the same on each component. Denote the direct product of these identical systems by T: X → X where X = [0, 1]L. From this product system we obtain a coupled map lattice (CML) Sε: X → X, if we introduce some interaction between the components, e.g. by averaging between nearest neighbours. The strength of the coupling depends upon some parameter ε.

For a broad class of piecewise expanding single-component-transformations τ we study such systems via their transfer operators and treat the coupled system as a perturbation of the uncoupled one. This yields existence and stability results for T-invariant measures with absolutely continuous finite-dimensional marginals.
A condition for unique ergodicity of minimal symbolic flows
Authors: Boshernitzan Michael D.
Published: September 2008

A sufficient condition for unique ergodicity of symbolic flows is provided. In an important but special case of interval exchange transformations, the condition has already been validated by W. Veech.
Pascal's triangle, dynamical systems and attractors
Authors: Haeseler Fritz V. , Peitgen Heinz-Otto , Skordev Gencho
Published: September 2008

This paper establishes a global dynamical systems approach for the fractal patterns which are obtained when analysing the divisibility of binomial coefficients modulo a prime power. The general framework is within the class of hierarchical iterated function systems. As a consequence we obtain a complete deciphering of the hierarchical self-similarity features.
The Wiener—Wintner property for the helical transform of the shift on [0, 1]ℤ
Authors: Assani I.
Published: September 2008

Let ([0, 1]z, ℬ([0, 1]Z), μz, ϕ) be the dynamical system where ϕ is the shift on the product of the unit interval with Lebesgue measure. We show that this dynamical system has the following properties: (1) there exists ƒ ∈ L1(μz) (in fact in L(Log Log L)β) where 0 <β< 1 for which the following Wiener-Wintner property does not hold: (W-W). There exists a single null set N ⊂ X off which for all x ∈ X/N the sequence converges for all ε ⊂ [0,1).

(2) The property (W-W) holds in all Lp(μz), 1< p≤∞. Added to a continuity property of the helical transform, (W-W) is equivalent to the Carleson-Hunt result on the pointwise convergence of Fourier series.
Existence de points fixes enlacés à une orbite périodique d'un homéomorphisme du plan
Authors: Bonatti Christian , Kolev Boris
Published: September 2008

Let ƒ be an orientation-preserving homeomorphism of the plane such that ƒ-Id is contracting. Under these hypotheses, we establish the existence for every periodic orbit , of a fixed point which has a nonzero linking number with .
Rokhlin towers and closing for flows on T2
Authors: Carroll C. R.
Published: September 2008

This paper considers theclosing problem,r≥2, for flows on T2with finitely many singularities. We study the action of such flows on circles transverse to the flow, using Katznelson and Ornstein's picture of rigid rotations as Rokhlin towers. We are able to obtain closing on purely topological grounds for a large class of flows by means of perturbations which simply twist a flow along an embedded transverse circle (twist-perturbations). However, we give an example of a flow (with finitely many singularities) for which nosmall twist-perturbation yields closing; indeed such perturbations either leave the non-wandering set unchanged or else collapse it to the set of singularities.
Commuting endomorphisms of the circle
Authors: Johnson Aimee S. A. , Rudolph Daniel J.
Published: September 2008

In this paper the results of Shub and Sacksteder are extended to the following theorem: let ƒ1 and ƒ2 be two commuting, expansive, orientation-preserving maps of the circle with a common fixed point and with both in C1+ε or Cr, r ≥2. Assume ƒ1 is p-to-1 and ƒ2 is q-to-1 where p and q generate a nonlacunary semigroup. Then there exists a diffeomorphism g of the same class such that gƒ1g−1 = TP and gƒ2g−1 = Tq.
On dynamics of triangular maps of the square
Authors: Kolyada S. F.
Published: September 2008

The paper is devoted to the triangular maps of the square into itself. The results presented were recently obtained by the author and are briefly stated (in Russian) in a difficult paper as well as those (jointly published with A. N. Sharkovsky) published in ECIT-89 (abstract). All these results are systematized and extended by the new ones. The more detailed proofs of all statements are given. It is shown, for example, that triangular maps exist such that their minimal attraction centres do not coincide with the centres, as well as such ones exist that the Milnor attractor is not contained in the closure of the set of periodic points.
Coalescence of circle extensions of measure-preserving transformations
Authors: Lemańczyk Mariusz , Liardet Pierre , Thouvenot Jean-Paul
Published: September 2008

We prove that for each ergodic automorphism T:(X, ℬ, μ)→(X, ℬ, μ) for which we can find an element S∈C(T) such that the corresponding Z2-action (S, T) on (X, ℬ, μ) is free, there exists a circle valued cocycle φ such that the group extension Tφ is ergodic but is not coalescent. In particular, the existence of such a cocycle is proved for all ergodic rigid automorphisms. As a corollary, in the class of ergodic transformations of [0,1) × [0,1) given by

for each irrational α we find φ such that Tφ is not coalescent. In some special cases the group law of the centralizer is given.
Characterizations of topological dynamical systems whose transformation group C*-algebras are antiliminal and of type 1
Authors: Aoki Nobuo , Tomiyama Jun
Published: September 2008

For a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.
Sinai–Ruelle–Bowen measures for N-dimensional derived from Anosov diffeomorphisms
Authors: Carvalho Maria
Published: September 2008

This paper is about the existence of transitive non-hyperbolic attractors with corresponding SRB measures for arcs of diffeomorphisms crossing the boundary of the Axiom A systems, obtained through an elementary generic bifurcation (Hopf, saddle-node or flip) on a transitive Anosov diffeomorphism or an attracting basic set.
Integer Cantor sets and an order-two ergodic theorem
Authors: Fisher Albert M.
Published: September 2008

Let denote the orbit closure, under the left shift σ, of the sequence… (all zeroes)… 101000101000000000101 … corresponding to the integer Cantor set . We prove that with respect to the infinite invariant measure ρ, which is the unique normalized non-atomic invariant measure on M, for every f ∈ L1(M, ρ), for ρ-a.e. x∈ M

where d = log 2/log 3, and c is the almost-sure value of the right-hand order-two density of the middle-third Cantor set. The proof uses renormalization to a scaling flow, plus identification of (M, σ) as a tower over the Kakutani-von Neumann dyadic odometer.
Some ergodic properties of commuting diffeomorphisms
Authors: Hu Huyi
Published: September 2008

For a smooth ℤ2-action on a C∞ compact Riemannian manifold M, we discuss its ergodic properties which include the decomposition of the tangent space of M into subspaces related to Lyapunov exponents, the existence of Lyapunov charts, and the subadditivity of entropies.
An estimate of entropy for toroidal chaos
Authors: Kwapisz Jaroslaw
Published: September 2008

For a mapping F:ℝ2→ℝ2 being a lift of an isotopic to the identity homeomorphism of the two-dimensional torus the rotation set ρ(F) consists of limit points of all sequences where xi, ∈ ℝ2 and ni→∞. It is known that if ρ(F) has nonempty interior then h(f) (the topological entropy of ƒ) is positive. We provide an estimate from below of h(f) in terms of ρ(F).
Newton's method and a class of meromorphic functions without wandering domains
Authors: Bergweiler Walter
Published: September 2008

LetNbe the class of meromorphic functionsfwith the following properties:fhas finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points offare zeros off′ and vice versa, with finitely many exceptions;fhas finite order. It is proved that iff∈N, thenfdoes not have wandering domains. Moreover, iff∈Nand if ∞ is among the limit functions offnin a cycle of periodic domains, then this cycle contains a singularity off−1. (Herefndenotes thenth iterate off) These results are applied to study Newton's method for entire functionsgof the formwherepandqare polynomials and wherecis a constant. In this case, the Newton iteration functionf(z) =z−g(z)/g′(z) is inN. It follows thatfn(z) converges to zeros ofgfor allzin the Fatou set off, if this is the case for all zeroszofg″. Some of the results can be extended to the relaxed Newton method.
Amenable actions of discrete groups
Authors: Elliott G. A. , Giordano T.
Published: September 2008

A structure theorem is established for amenable actions of a countable discrete group.
Lefschetz formulae for Anosov flows on 3-manifolds
Authors: Sánchez-Morgado Héctor
Published: September 2008

Fried has related closed orbits of the geodesic flow of a surface S of constant negative curvature to the R-torsion for a unitary representation of the fundamental group of the unit tangent bundle T1S. In this paper we extend those results to transitive Anosov flows and 2-dimensional attractors on 3-manifolds.
Connectedness of the tricorn
Authors: Nakane Shizuo
Published: September 2008

In this note, we show the connectedness of the tricorn, the connectedness locus for the family of antiquadratic maps: fc(z) = + c, c ∈ C.
Symplectic displacement energy for Lagrangian submanifolds
Authors: Polterovich Leonid
Published: September 2008

Recently H. Hofer defined a new symplectic invariant which has a beautiful dynamical meaning. In the present paper we study this invariant for Lagrangian submanifolds of symplectic manifolds. Our approach is based on Gromov's theory of pseudo-holomorphic curves.
The crossed product of a UHF algebra by a shift
Authors: Bratteli Ola , Størmer Erling , Kishimoto Akitaka , Rørdam Mikael
Published: September 2008

We prove that the crossed product of the CAR algebra M2∞ by the shift is an inductive limit of homogeneous algebras over the circle with fibres full matrix algebras. As a consequence the crossed product has real rank zero, and where is the Cuntz algebra of order 2.
Knaster-like continua and complex dynamics
Authors: Devaney Robert L.
Published: September 2008

In this paper we discuss the topology and dynamics ofEλ(z) = λezwhen λ is real and λ > 1/e. It is known that the Julia set ofEλis the entire plane in this case. Our goal is to show that there are certain natural invariant subsets forEλwhich are topologically Knaster-like continua. Moreover, the dynamical behavior on these invariant sets is quite tame. We show that the only trivial kinds of α- and ω-limit sets are possible.
Global dynamical properties of Euler and backward Euler
Authors: Hockett Kevin
Published: September 2008

We investigate the global dynamics of both the Euler and backward Euler algorithms as applied to polynomial vector fields on the Riemann sphere = ℂ ∪ {∞}. The geometry and dynamics of solutions at infinity is seen to play an important role in organizing the global dynamics of the flow, and we show how to modify both algorithms to account for this structure, thereby obtaining global approximations to the flow.
Mixing sets and relative entropies for higher-dimensional Markov shifts
Authors: Kitchens Bruce , Schmidt Klaus
Published: September 2008

We consider certain measurable isomorphism invariants for measure-preserving ℤd-actions on probability spaces, compute them for a class of d-dimensional Markov shifts, and use them to prove that some of these examples are non-isomorphic. The invariants under discussion are of three kinds: the first is associated with the higher-order mixing behaviour of the ℤd-action, and is related—in this class of examples—to an an arithmetical result by David Masser, the second arises from certain relative entropies associated with the ℤd-action, and the third is a collection of canonical invariant σ-algebras. The results of this paper are generalizations of earlier results by Kitchens and Schmidt, and we include a proof of David Masser's unpublished theorem.
Multiphase averaging for generalized flows on manifolds
Authors: Dumas H. S. , Golse F. , Lochak P.
Published: September 2008

We present a new proof of a strengthened version of Anosov's multiphase averaging theorem, originally stated for systems of ODEs with slow variables evolving in Rm and fast variables evolving on a smooth immersed manifold. Our result allows the fast variables to belong to an arbitrary smooth compact Riemannian manifold, and the vector field to have only Sobolev regularity. This is accomplished using normal form techniques adapted to a slightly generalized version of the DiPema-Lions theory of generalized flows for ODEs.
On the existence of homoclinic orbits on Riemannian manifolds
Authors: Giannoni Fabio
Published: September 2008

We prove the existence of a non-trivial homoclinic orbit on a Riemannian manifold (possibly non-compact), for Hamiltonian systems of the second order of the form:

where the potential V is T-periodic in the time variable.
On the class of multipliers for W⊥
Authors: Glasner Eli
Published: September 2008

Let W⊥ be the class of all ergodic measure-preserving transformations (systems), which are disjoint from every weakly mixing system. Let M(W⊥) be the class of multipliers for W⊥; i.e. the class of all systems (X, μ, T) in W⊥ such that for every system (Y, ν, T) ∈ W⊥ and every ergodic joining λ of X and Y, the system (X × Y, λ, T × T) is also in W⊥. Well known results on disjointness show that the class D of ergodic distal systems, is a subclass of M(W⊥). Thus one has D ⊂ M(W⊥) ⊂ W⊥. Glasner and Weiss have shown that D ≠ W⊥. The purpose of this paper is to also show that D ≠ M(W⊥). The question whether M(W⊥) = W⊥ remains open.
Ergodic theorems for random sets with density zero
Authors: Huang Yenkun
Published: September 2008

We generalize a result of Bourgain and a result of Huang. We also give a positive solution to A. Bellow's question: the a.e. convergence of the averages for σn = 1/n. On the other hand, we establish a sufficient and necessary condition for random sets in Z+ with asymptotic density zero which almost surely satisfy a mean ergodic theorem.
Topological rigidity of strong stable foliations for Cartan actions
Authors: Hurder Steven
Published: September 2008

We show that the strongest stable foliations associated with the generators of a Cartan action on a compact infra-nilmanifold are invaraint under topological conjugacy. This has the corollary that a Cartan action on a compact infra-nilmanifold with constant exponents is smoothly conjugate to an affine action.
Natural Lagrangian systems without conjugate points
Authors: Innami Nobuhiro
Published: September 2008

The variation vector fields through extremals of the variational principles of natural Lagrangian functions satisfy the equation of Jacobi type. By making use of the Jacobi equation we obtain the estimates of measure-theoretic entropy for natural Lagrangian systems without conjugate points.
Distortion results and invariant Cantor sets of unimodal maps
Authors: Martens Marco
Published: September 2008

A distortion theory is developed for S-unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of S-unimodal maps is classified according to a distortion property, called the Markov-property.
Properties of invariant measures for piecewise expanding one-dimensional transformations with summable oscillations of derivative
Authors: Góra Pawel
Published: September 2008

We study the properties of absolutely continuous invariant measures for one-dimensional transformations satisfying Schmitt's condition. We prove existence, constrictiveness of the induced Perron-Frobenius operator, openness of supports, estimate on a number of ergodic components, exponential decay of correlation, Bernoulli property, etc.
Convergence and divergence of ergodic averages
Authors: Jones Roger L. , Wierdl Mate
Published: September 2008

In this paper we consider almost everywhere convergence and divergence properties of various ergodic averages. A general method is given which can be used to construct averages for which a.e. convergence fails, and to show divergence (and in some cases ‘strong sweeping out’) for large classes of ergodic averages. We also show that there are sequences with the gaps between successive terms converging to zero, but such that the Cesaro averages obtained by sampling a flow along these sequences of times converge a.e. for all f∈L1(X).
Flow equivalence of reducible shifts of finite type
Authors: Huang Danrung
Published: September 2008

Using an invariant of Cuntz, we classify reducible shifts of finite type with two irreducible components up to flow equivalence.
Symbolic dynamics for angle-doubling on the circle III. Sturmian sequences and the quadratic map
Authors: Keller Karsten
Published: September 2008

By the theory of Douady and Hubbard, the structure of Julia sets of quadratic maps is tightly connected with the angle-doubling maphon the circleT. In particular, a connected and locally connected Julia set can be considered as a topological factorT/ ≈ ofTwith respect to a specialh-invariant equivalence relation ≈ onT, which is called Julia equivalence by Keller. Following an idea of Thurston, Bandt and Keller have investigated a map α →αfromTonto the set of all Julia equivalences, which gives a natural abstract description of the Mandelbrot set. By the use of a symbol sequence called the kneading sequence of the point α, they gave a topological classification of the abstract Julia setsT/α. It turns out thatT/αcontains simple closed curves iff the point α has a periodic kneading sequence. The present article characterizes the set of points possessing a periodic kneading sequence and discusses this set in relation to Julia sets and to the Mandelbrot set.
Heteroclinics for a reversible Hamiltonian system
Authors: Rabinowitz Paul H.
Published: September 2008

This paper uses an elementary variational argument to establish the existence of solutions heteroclinic to a pair of periodic orbits for a class of Hamiltonian systems including Hamiltonians of multiple pendulum type.
Nonexistence of SBR measures for some diffeomorphisms that are ‘Almost Anosov’
Authors: Hu Huyi , Young Lai-Sang
Published: September 2008

The purpose of this paper is to present some simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. Each example has a fixed point at which the larger eigenvalue is equal to one and the smaller eigenvalue is less than one.
Fibonacci maps re(aℓ)visited
Authors: Keller Gerhard , Nowicki Tomasz
Published: September 2008

We prove that unimodal Fibonacci maps with negative Schwarzian derivative and a critical point of order ℓ have a finite absolutely continuous invariant measure if ℓ ∈ (1 ℓ1) where ℓ1 is some number strictly greater than 2. This extends results of Lyubich and Milnor for the case ℓ = 2.
Ergodic theory for axiom A endomorphisms
Authors: Qian Min , Zhang Zhusheng
Published: September 2008

In this paper the Pesin's entropy formula and ‘large ergodic theorem’ are constructed for Axiom A endomorphisms.
Caractérisation des flots d' Anosov en dimension 3 par leurs feuilletages faibles
Authors: Barbot Thierry
Published: September 2008

We consider Anosov flows on closed 3-manifolds. We show that if such a flow admits a weak foliation whose lifting in the universal covering is a product foliation, thenit is characterized up to topological equivalence by its weak stable foliation up to topological conjugacy. As a corollary we obtain that, up to topological equivalence and finite coverings, suspensions and geodesic flows are the unique Anosov flows on closed 3-manifolds whose weak stable foliations are transversely projective.
Robust nonhyperbolic dynamics and heterodimensional cycles
Authors: Díaz Lorenzo J.
Published: September 2008

We describe an open set A of arcs of diffeomorphisms bifurcating through the creation of heterodimensional cycles for which every diffeomorphism after the bifurcation is nonhyperbolic or unstable. We also prove that generically in A the borning nonwandering set is transitive and local maximal for a full (Lebesgue) set of parameter values.
A note on transitive flows
Authors: Lawrence Deborah A. , Thomas Edward S.
Published: September 2008

In this article, we study transitive flows on manifolds or on manifold-like phase spaces. These are flows in which at least one orbit is dense. By quantifying the nature of the set of dense orbits, a variety of transitivity types is obtained. We explore conditions under which products or ‘scaled’ products of transitive flows are again transitive. We also investigate a connection between transitivity and topologically strong mixing.
Ruelle's transfer operator for random subshifts of finite type
Authors: Bogenschütz Thomas , Gundlach Volker Mathias
Published: September 2008

We consider a Ruelle—Perron—Frobenius type of selection procedure for probability measures that are invariant under random subshifts of finite type. In particular we prove that for a class of random functions this method leads to a unique probability exhibiting properties that justify the names Gibbs measure and equilibrium states. In order to do this we introduce the notion of bundle random dynamical systems and provide a theory for their invariant measures as well as give a precise definition of Gibbs measures.
The Hausdorff dimension of certain solenoids
Authors: Bothe H. G.
Published: September 2008

For the solid torus V = S1 × and a C1 embedding f: V → V given by with dϕ/dt > 1, 0 < λi(t) < 1 the attractor Λ = ∩i = 0∞fi(V) is a solenoid, and for each disk D(t) = {t} × (t ∈ S1) the intersection Λ(t) = Λ ∩ D(t) is a Cantor set. It is the aim of the paper to find conditions under which the Hausdorff dimension of Λ(t) is independent of t and determined by where the real numbers pi are characterized by the condition that the pressure of the function log : S1 → ℝ with respect to the expanding mapping ϕ: S1 → S1 becomes zero. (There are two further characterizations of these numbers.)

It is proved that (0.1) holds provided λ1, λ2 are sufficiently small and Λ satisfies a condition called intrinsic transverseness. Then it is shown that in the C1 space of all embeddings f with sup λi > Θ−2 (Θ the mapping degree of ϕ: S1 → S1) all those f which have an intrinsically transverse attractor Λ form an open and dense subset.
Rigidity for non-compact surfaces of finite area and certain Kähler manifolds
Authors: Cao Jianguo
Published: September 2008

We first consider the rigidity of the marked length spectrum for non-compact surfaces of finite area.
Common extensions and hyperbolic factor maps for coded systems
Authors: Fiebig Doris
Published: September 2008

The classification of dynamical systems by the existence of certain common extensions has been carried out very successfully in the class of shifts of finite type (‘finite equivalence’, ‘almost topological conjugacy‘). We consider generalizations of these notions in the class of coded systems. Topological entropy is shown to be a complete invariant for the existence of a common coded entropy preserving extension. In contrast to the shift of finite type setting, this extension cannot be made bounded-to-1 in general. Common extensions with hyperbolic factor maps lead to a version of almost topological conjugacy for coded systems.
A subshift of finite type that is equivalent to the Ising model
Authors: Häggström Olle
Published: September 2008

For the Ising model with rational parameters we show how to construct a subshift of finite type that is equivalent to this Ising model, in that the translation invariant Gibbs measures for the Ising model and the measures of maximal entropy for the subshift of finite type can be identified in a natural way. This is generalized to the non-translation invariant case as well. We also show how to construct, given any H > 0, an ergodic measure of maximal entropy for a subshift of finite type and a continuous factor, such that the factor has entropy H.
Higher cohomology for Abelian groups of toral automorphisms
Authors: Katok Anatole , Katok Svetlana
Published: September 2008

We give a complete description of smooth untwisted cohomology with coefficients in ℝl for ℤk-actions by hyperbolic automorphisms of a torus. For 1 ≤ n ≤ k − 1 the nth cohomology trivializes, i.e. every cocycle is cohomologous to a constant cocycle via a smooth coboundary. For n = k a counterpart of the classical Livshitz Theorem holds: the cohomology class of a smooth k-cocycle is determined by periodic data.
Entropy pairs for a measure
Authors: Blanchard F. , Host B. , Maass A. , Martinez S. , Rudolph D. J.
Published: September 2008

We define entropy pairs for an invariant measure µ on a topological dynamical system (X, T), and show they allow one to construct the maximal topological factorwith entropy 0 for µ. Then we prove that for any µ, a µ-entropy pair is always topologically so, and the reverse is true when (X, T) is uniquely ergodic.
Instances of cohomological triviality and rigidity
Authors: Parry William
Published: September 2008

We consider a compact metric zero-dimensional abelian group X and mixing ℤ2 (or ℤ2+) actions by automorphisms (or endomorphisms). We provide various instances of cohomological triviality and rigidity for cocycles taking values in finite groups.
Classifying C1+ structures on dynamical fractals: 1. The moduli space of solenoid functions for Markov maps on train tracks
Authors: Pinto A. A. , Rand D. A.
Published: September 2008

Sullivan's scaling function provides a complete description of the smooth conjugacy classes of cookie-cutters. However, for smooth conjugacy classes of Markov maps on a train track, such as expanding circle maps and train track mappings induced by pseudo-Anosov systems, the generalisation of the scaling function suffers from a deficiency. It is difficult to characterise the structure of the set of those scaling functions which correspond to smooth mappings. We introduce a new invariant for Markov maps called the solenoid function. We prove that for any prescribed topological structure, there is a one-to-one correspondence between smooth conjugacy classes of smooth Markov maps and pseudo-Hölder solenoid functions. This gives a characterisation of the moduli space for smooth conjugacy classes of smooth Markov maps. For smooth expanding maps of the circle with degree d this moduli space is the space of Hölder continuous functions on the space {0,…, d − 1}ℕ satisfying the matching condition.
Cohomological rigidity of algebraic ℤd-actions
Authors: Schmidt K.
Published: September 2008

We prove that every Hölder-continuous cocycle for an expansive and mixing ℤd-action α by automorphisms of a compact, abelian group X is Hölder-cohomologous to an affine cocycle, and give a complete description of the Hölder cohomology group of α.
Absolutely continuous invariant measures for piecewise expanding C2 transformations in Rn on domains with cusps on the boundaries
Authors: Adl-Zarabi Kourosh
Published: September 2008

Let Ω be a bounded region in Rn and let be a partition of Ω into a finite number of subsets having piecewise C2 boundaries. The boundaries may contain cusps. Let τ: Ω → Ω be piecewise C2 on and expanding in the sense that there exists α > 1 such that for any i = 1, 2,…,m, where is the derivative matrix of and ‖·‖ is the euclidean matrix norm. The main result provides a lower bound on α which guarantees the existence of an absolutely continuous invariant measure for τ.
Nonuniformly hyperbolic K-systems are Bernoulli
Authors: Chernov N. I. , Haskell C.
Published: September 2008

We prove that those non-uniformly hyperbolic maps and flows (with singularities) that enjoy the K-property are also Bernoulli. In particular, many billiard systems, including those systems of hard balls and stadia that have the K-property, and hyperbolic billiards, such as the Lorentz gas in any dimension, are Bernoulli. We obtain the Bernoulli property for both the billiard flows and the associated maps on the boundary of the phase space.
Some results on the classification of expansive automorphisms of compact abelian groups
Authors: Fagnani Fabio
Published: September 2008

In this paper we study expansive automorphisms of compact 0-dimensional abelian groups. Our main result is the complete algebraic and topological classification of the transitive expansive automorpisms for which the maximal order of the elements isp2for a primep. This yields a classification of the transitive expansive automorphisms with topological entropy logp2. Finally, we prove a necessary and sufficient condition for an expansive automorphism to be conjugated, topologically and algebraically, to a shift over a finite group.
Construction of invariant measures supported within the gaps of Aubry–Mather sets
Authors: Forni Giovanni
Published: September 2008

This paper represents a contribution to the variational approach to the understanding of the dynamics of exact area-preserving monotone twist maps of the annulus, currently known as the Aubry–Mather theory. The method introduced by Mather to construct invariant measures of Denjoy type is extended to produce almost-periodic measures, having arbitrary rationally independent frequencies, and positive entropy measures, supported within the gaps of Aubry–Mather sets which do not lie on invariant curves. This extension is based on a generalized version of the Percival's Lagrangian and on a new minimization procedure, which also gives a simplified proof of the basic existence theorem for the Aubry–Mather sets.
On the transfer operator for rational functions on the Riemann sphere
Authors: Denker Manfred , Przytycki Feliks , Urbański Mariusz
Published: September 2008

Let T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T−n(B(x, r)), centered at any point x in its Julia set J = J(T), does not exceed Lnrp for some constants L ≥ 1 and ρ > 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) > supz∈Jφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.
θ-transformations, θ-shifts and limit theorems for some Riesz-Raikov sums
Authors: Petit Bernard
Published: September 2008

By Fourier methods, it is possible to prove central limit theorems for Riesz-Raikov sums for any real θ > 1 and for a quite large class of functions f.

The same problem is solved here for Hölder-continuous functions f and Pisot–Vijayaragavan numbers θ in a totally different way: it is shown that the question is equivalent to working with ergodic sums for suitable functions F on [0;l], and T denoting the θ-transformation x ↦ θx mod 1. In addition, limit theorems are proved on the θ-shifts, for any real θ 1.
Simple C*-crossed products with a unique trace
Authors: Bédos Erik
Published: September 2008

Let (α, u) denote a cocycle-crossed action of a discrete group G on a unital C*-algebra A, and B = C*r (A, G, α, u) the associated reduced (twisted) C*-crossed product. We discuss the following problem: when is B a simple C*-algebra with a unique trace? To this aim, we introduce the concept of tracially properly outerness of actions.
The lap-counting function for linear mod one transformations I: explicit formulas and renormalizability
Authors: Flatto Leopold , Lagarias Jeffrey C.
Published: September 2008

Linear mod one transformations are the maps of the unit interval given by fβα(x) = βx + α (mod 1), with β > 1 and 0 ≤ α < 1. The lap-counting function is the function where the lap number Ln essentially counts the number of monotonic pieces of the nth iterate . We derive an explicit factorization formula for Lβα(z) which directly shows that Lβα(z) is a function meromorphic in the open unit disk {z: |z| < 1} and analytic in the open disk {z: |z| < 1/β}, with a simple pole at z = 1/β.Comparison with a known formula for the Artin—Mazur—Ruelle zeta function ζβ,α(z) of fβα shows that Lβα(z) and ζβ,α(z) have identical sets of singularities in the disk {z: |z| < 1}. We derive two more factorization formulae for Lβ,α(z) valid for certain parameter ranges of (β, α). When 1 < α + β ≤ 2, there is sometimes a ‘renormalization’ structure of such maps present, which has previously been studied in connection with simplified models for the Lorenz attractor. In the case that fβα is non-trivially renormalizable, we obtain a factorization formula for Lβα(z). For (β, α) in a region contained in 2 < α + β ≤ 3 we obtain a factorization formula which relates Lβα(z) to a ‘rescaled’ lap-counting function from the region 1 < α + β ≤ 2. The various factorizations exhibit certain singularities of Lβα(z) on the circle |z| = 1/β. These singularities are related to topological dynamical properties of fβ,α. In parts II and III we show that these comprise the complete set of such singularities on the circle |z| = 1/β.
Automorphisms of the Bernoulli endomorphism and a class of skew-products
Authors: Parry William
Published: September 2008

We consider a class of skew-products with base the one-sided two-shift equipped with the (½, ½)Bernoulli measure and fibre either or the circle. We give conditions for the first kind to be isomorphic to the base itself and use this result to establish an isomorphism with the base for the second kind when a certain irrationalis very well approximated by rationals. A consequence is that there is a circle action of automorphisms commuting with theone-sided two-shift.
Non-ergodicity for C1 expanding maps and g-measures
Authors: Quasf Anthony N.
Published: September 2008

We introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.
Intermingled basins for the triangle map
Authors: Alexander James C. , Hunt Brian R. , Kan Ittai , Yorke James A.
Published: September 2008

A family of quadratic maps of the plane has been found numerically for certain parameter values to have three attractors, in a triangular pattern, with ‘intermingled’ basins. This means that for every open set S, if the basin of attraction of one of the attractors intersects S in a set of positive Lebesgue measure, then so do the other two basins. In this paper we mathematically verify this observation for a particular parameter, and prove that our results hold for a set of parameters with positive Lebesgue measure.
Rank and symbolic complexity
Authors: Ferenczi Sébastien
Published: September 2008

We investigate the relation between the complexity function of a sequence, that is the number p(n) of its factors of length n, and the rank of the associated dynamical system, that is the number of Rokhlin towers required to approximate it. We prove that if the rank is one, then lim , but give examples with lim for any prescribed function G with G (n) = 0(an) for every a > 1. We give exact computations for examples of the ‘staircase’ type, which are strongly mixing systems with quadratic complexity. Conversely, for minimal sequences, if p(n) < an + b for some a ≥ 1, the rank is at most 2[a], with bounded strings of spacers, and the system is generated by a finite number of substitutions.
Ergodic elements for actions of Lie groups
Authors: Gutschera K. Robert
Published: September 2008

Given a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.