The structure of the cohomology equivalence classes for non-singular, not necessarily invertible mappings of probability spaces is studied. In particular, some results of Kochergin and Ornstein–Smorodinsky on the structure of these classes for measure-preserving automorphisms are generalized to the case of non-singular endomorphisms. Our approach is based on Hopf's maximal ergodic theorem and its proof by Garsia.

Let $A\in SL(2,{\Bbb Z})$ be given and $n\in{\Bbb N}$. $A$ induces a map $A_n$ on ${\Bbb Z}^2_n$ by $(x,y)\mapsto(a_{11}x+a_{12}y,a_{21}x+a_{22}y)$ mod $n$. $A_n$ can be thought of as a discrete approximation of the similarly defined map on the unit square. It is known that the $A_n$ have a surprisingly small period: there always is an $r\le3n$ such that $A^r_n$ is the identity on ${\Bbb Z}^2_n$.

One can visualize the properties of the iterates of $A_n$ by considering the sets $P,A_nP,A^2_nP,\ldots$ for a fixed subset $P$ of ${\Bbb Z}^2_n$. (Here the cat comes into play: $P$ has often been chosen to be the digitalized picture of a cat, in particular when treating the special case of the Anosov map $A={2\ 1\atopwithdelims() 1\ 1}$.) It turns out that in the sequence $(A^k_nP)_k$ mostly ‘chaotic’ pictures occur as one would expect. However, some kind of regular behaviour can also be observed for certain powers $A^k_n$ before the end of the period.

It is the aim of this paper to describe these phenomena and to explain why and how they occur. In order to quantify the chaotic behaviour of the maps on ${\Bbb Z}^2_n$ under consideration, we introduce suitable functions.

Our methods are elementary. Only basic results from the geometry of numbers will be used.

In this paper we show that some nonsimply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, section I.5]. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular, we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.

We study some diffeomorphisms in the boundary of the set of Anosov diffeomorphisms mainly from the ergodic viewpoint. We prove that these diffeomorphisms, obtained by isotopy from an Anosov $f:M \mapsto M$ through a heteroclinic tangency, determine a manifold ${\cal M}$ of finite codimension in the set of $C^r$ diffeomorphisms. We prove that any diffeomorphism $F$ in ${\cal M}$ is conjugate to $f$; moreover, there exists a unique SRB measure for $F$, and $F$ is Bernoulli with respect to this measure. In particular, if the dimension of $M$ is two, and $\mu $ is a volume element, we prove that the isotopy can be taken such that the measure is preserved.

Integer $m\times m$ matrices $A$ with determinant $1$ define diffeomorphisms of the $m$-dimensional torus $T^m=({\Bbb R}/{\Bbb Z})^m$ into itself. Likewise, they define bijective self-maps of the discretized tori $({\Bbb Z}/n{\Bbb Z})^m=({\Bbb z}_n)^m$. We present estimates of the surprisingly low order (or period) $\Per_A(n)$ of the iteration $A^r, r=1,2,3,\ldots,$ on the discretized torus $({\Bbb z}_n)^m$. We obtain $\Per_A(n)\leq 3n$ for dimension $m=2$. In the special case of the Anosov map $A={2\ 1 \atopwithdelims() 1\ 1}$, this result is due to Dyson and Talk [DT92]. For arbitrary dimensions $m>2$ we obtain $$\Per_A(n)\leq {\rm constant}\cdot n^{m-1},$$ provided $n$ is a power of a prime number. For general $n$, number theoretic problems arise.

Let $\{w_i,p_i\}$ be a contractive iterated function system (IFS) [1, pp. 79–80] with probabilities, i.e. a set of contraction maps $w_i:X\to X$ with associated probabilities $p_i$, $i=1,2,\ldots,N$. We provide a simple proof that for almost every address sequence $\sigma$ and for all $x$ the limit $\lim_{n\to \infty}1/n\sum_{i\le n}f(w_{\sigma_n}\circ w_{\sigma_{n-1}}\circ\cdots\circ w_{\sigma_1}(x))$ exists and is equal to $\int_Xf(z)\,d\mu(z)$, where $\mu$ is the invariant measure of the IFS. This is the so called ‘ergodic property’ for the IFS and was proved by Elton in [3]. However, the uniqueness of the invariant measure was not previously exploited. This provides considerable simplification to the proof.

A conjugacy between two uniformly asymptotic affine circle expanding maps is symmetric if they have the same eigenvalues along corresponding periodic points.

In this paper, the structure of $r$ equivalence, which was introduced by Vershik and which classifies group actions of the group $G=\sum_{n=1}^\infty{\Bbb Z}\slash r_n{\Bbb Z}$, $r_n\in{\Bbb N}\setminus\{1\}$, is examined. This is an equivalence relation that naturally arises from looking at certain sequences of $\sigma$-algebras. Vershik proved that if a sequence $r=(r_1,r_2,\ldots)$ does not satisfy a super-rapid growth rate, then entropy is an invariant for $r$ equivalence. In this paper, a strong converse of this is proven: for any $r$ which does satisfy this super-rapid growth rate, we can find a zero entropy action in every $r$ equivalence class.

For every Pisot number $\theta$, and every real number $\beta$ greater than one not equivalent to $\theta$, if $\mu$ is a Bernoulli measure relative to the $\theta$-transformation, then for $\mu$-almost all $x$ the sequence $(x\beta^n)_{n\geq 0}$ is uniformly distributed modulo one.

We show that for a large class of piecewise monotonic transformations on a totally ordered, compact set one can construct conformal measures and obtain the exponential mixing rate for the associated equilibrium state. The method is based on the study of the Perron–Frobenius operator. The conformal measure, the density of the invariant measure and the rate of mixing are deduced by using an appropriate Hilbert metric, without any compactness arguments, even in the case of a countable to one transformation.

Let $F:X\rightarrow X$ be a $C^k(X)$, $k=[0, \infty]$, map on a topological space (smooth manifold) $X$, $A:X\rightarrow \End(C^m)$ and let $\{U_\alpha\}$ be an $F\mbox{-invariant}$ covering of $X$. We introduce spaces of cohomologies associated with $\{U_\alpha\}$ and an operator $T=I - R$, where $(R\phi)(x)=A(x)\phi(F(x))$ is a weighted substitution operator in $C^k(X)$. This yields a correspondence between $\Im T$ and $\Im T|U_\alpha$ and the description of $\Im T$ in cohomological terms. In particular, it is proven that for any structurally stable diffeomorphism on a circle and for large enough $k$, the operator $T$ is semi-Fredholm, and a similar result holds for the substitution operators generated by simple multidimensional maps. On the other hand, we show that, in general, the closures of $\Im T$ and $\Im T|U_\alpha$ are independent.

Techniques from the authors' previous paper [BD1] are applied to unique $G$-measures, where the $g$-functions are uniformly Lipschitz. Conditions are found when such a measure is, and is not, equivalent to a measure of product type.

We define rank one for ${\Bbb Z}^d$ actions and show that those rank one actions with a certain tower shape are loosely Bernoulli for $d\ge 1$. We also construct a zero entropy ${\Bbb Z}^2$ loosely Bernoulli action with a zero entropy, ergodic, non-loosely Bernoulli one-dimensional subaction.

A new approach to the study of the rate of mixing in Anosov flows, recently proposed by Chernov, is simplified and generalized to the higher-dimensional case.

We give a formula for the sum of the Lyapunov exponents of a nondegenerate polynomial map $f$ of ${\bf C}^{2}$ close to $(z,w)\to (z^d,w^d)$. The formula only involves the behavior of $f$ at infinity. In particular, it follows that the sum only depends on the homogeneous part of $f$ of degree $d$.

Let $\Gamma$ denote a word-hyperbolic group, and let $S=S^{-1}$ denote a finite symmetric set of generators. Let $S_n = \{w: |w| = n \}$ denote the sphere of radius $n$, where $|\cdot|$ denotes the word length on $\Gamma$ induced by $S$. Define $\s_n \deq (1/{\#S_n}) \sum_{w \in S_n}w$, and $\mu_n=(1/(n+1))\sum_{k=0}^n \s_k$. Let $(X,{\cal B},m)$ be a probability space on which $\Gamma$ acts ergodically by measure-preserving transformations. We prove a strong maximal inequality in $L^2$ for the maximal operator $f_{\mu}^*=\sup_{n\ge 0}|\mu_n f(x)|$. The maximal inequality is applied to prove a pointwise ergodic theorem in $L^2$ for exponentially mixing actions of $\Gamma$ of the following form: $\mu_n f(x) \to \int_X f\,dm$ almost everywhere and in the $L^2$-norm, for every $f\in L^2(X)$. As a corollary, for a uniform lattice $\Gamma \subset G$, where $G$ is a simple Lie group of real rank one, we obtain a pointwise ergodic theorem for the action of $\Gamma$ on an arbitrary ergodic $G$-space. In particular, this result holds when $X=G/\Lambda$ is a compact homogeneous space, and yields an equidistribution result for sets of lattice points of the form $\Gamma g$, for almost every $g\in G$.

We show here that some finite measure on $[-\pi,\pi]$ can never be obtained as a spectral measure of a transformation induced by a rotation. For this, we propose a new way to build a Kronecker set, which leads to a non loosely Bernoulli Gaussian–Kronecker automorphism.

On montre ici qu'une certaine mesure finie sur $[-\pi,\pi]$ ne peut jamais être obtenue comme mesure spectrale d'une transformation induite par une rotation. On propose pour cela une nouvelle façon de construire un ensemble de Kronecker, qui permet de voir que certains systèmes dynamiques gaussiens–Kronecker ne sont pas lâchement Bernoulli.

Let $F$ be a lift of a homeomorphism $f: {\Bbb T}^{2} \to {\Bbb T}^{2}$ homotopic to the identity. We assume that the rotation set $\rho(F)$ is a line segment with irrational slope. In this paper we use the fact that ${\Bbb T}^2$ is necessarily chain transitive under $f$ if $f$ has no periodic points to show that if $v \in \rho(F)$ is a rational point, then there is a periodic point $x \in {\Bbb T}^{2}$ such that $v$ is its rotation vector.

For each irreducible hyperbolic automorphism $A$ of the $n$-torus we construct a sofic system $(\Sigma,\sigma)$ and a bounded-to-one continuous semiconjugacy from $(\Sigma,\sigma)$ to $({\Bbb T}^n,A)$. This construction is natural in the sense that it depends only on the characteristic polynomial of $A$ and, furthermore, it has an arithmetic interpretation.

We consider the entropy of systems of random transformations, where the transformations are chosen from a set of generators of a ${\Bbb Z}^d $ action. We show that the classical definition gives unsatisfactory entropy results in the higher-dimensional case, i.e. when $d \geq 2$. We propose a definition of the entropy for random group actions which agrees with the classical definition in the one-dimensional case, and which gives satisfactory results in higher dimensions. This definition is based on the fibre entropy of a certain skew product. We identify the entropy by an explicit formula which makes it possible to compute the entropy in certain cases.