A rational map $T$ of degree not less than two is known to preserve a measure, called the conformal measure, equivalent to the Hausdorff measure of the same dimension as its Julia set $J$ and supported there, with respect to which it is ergodic and even exact. As a consequence of Birkhoff's pointwise ergodic theorem almost every $z$ in $J$ with respect to the conformal measure has an orbit that is asymptotically distributed on $J$ with respect to this measure. As a counterpoint to this, the following result is established in this paper. Let $\Omega(z)=\Omega_{T}(z)$ denote the closure of the set $\{T^{n}(z):n=1,2,\ldots\}$. For any expanding rational map $T$ of degree at least two we set \[ S(z_{0})=\{z\in J:z_{0}\not\in \Omega_{T}(z)\}. \] We show that for all $z_{0}$ the Hausdorff dimensions of $S(z)$ and $J$ are equal.

We completely classify one-sided Markov chains up to measure-theoretic isomorphism. The classification is effective and computable.

A graphic flow is a totally minimal flow such that the only minimal subsets of the product flow are the graphs of the powers of the defining homeomorphism [2]. We consider flows of the form $(X^{k},T^{L})$, where $(X,T)$ is graphic, $k$ is a positive integer, and $L:\{1,\ldots,k\}\to {\Bbb Z}\setminus \{0\}$. It is shown that the isomorphism classes of these flows are determined by the cardinality of $L^{-1}(p)$.

In this paper we indicate how techniques developed by Zimmer [8] regarding certain smooth actions of discrete groups on compact manifolds can be extended to actions of discrete groups and semisimple Lie groups on compact fiber bundles. The actions that we consider preserve a smooth volume density on the total space of the bundle that defines a smooth field of volume densities on the fibers. When the standard fiber has ‘low’ dimension or the action preserves certain geometric structures on the fibers, Zimmer's techniques can show that the action must be measurably conjugate to a topologically relatively isometric action.

We find analytic solutions $\varphi_{t}$ to the conjugation equation $$ f_{t} \circ \varphi_{t}=\varphi_{t} \circ f_{0}, $$ where $f_{t}$ and $\varphi_{t}$ are analytic diffeomorphisms from ${\Bbb T}^{n}$ to itself that depend analytically on $t$, and $f_{0}$ is an Anosov toral diffeomorphism with reduced spectrum and which is diagonalizable.

We study a qualitative theory of compact, transversely piecewise-smooth foliated $S^1$-bundles. We show that it has the same qualitative properties as that of smooth codimension-one foliations on a compact manifold. By the obtained qualitative properties, we can deduce that the expansion growth of this foliation is classified and that the entropy of this foliation is positive if and only if there exists a resilient leaf.

We introduce transition entropy and periodic entropy for locally compact subshifts. Finiteness of both characterizes the existence of a finite generator. Finiteness of the transition entropy characterizes the existence of a generator with bounded degree. We extend Krieger's embedding theorem and characterize the locally compact non-dense subsystems of a (compact) mixing shift of finite type.

Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic in the unit disk $\vert z\vert <1$ and analytic in $\vert z\vert<1/\beta$, and part II showed that the singularities of $L_{\beta,\alpha}(z)$ on the circle $\vert z\vert=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$, where $N_{\beta,\alpha}$ is the period of the ergodic part of a Markov chain associated to $f_{\beta,\alpha}$. This paper proves that the set of singularities on $\vert z\vert=1/\beta$ is identical to the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$. Part II showed that $N_{\beta,\alpha}=1$ for $\beta> 2$, and this paper determines $N_{\beta,\alpha}$ in the remaining cases where $1<\beta\le 2$.

We show that for translation invariant Markov random fields: (1) the K-property implies a trivial full tail; (2) the Bernoulli property implies Følner independence. The existence of bilaterally deterministic Bernoulli shifts tells us that neither result is true without the Markov assumption (even in one dimension). We also show that for general translation invariant random fields: (3) Følner independence implies a trivial full tail.

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [9]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [16]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [16].

Nous développons dans cet article des résultats de M. B. Tabanov et moi-même annoncés dans la note [6]. M. B. Tabanov a donné un exposé de ses méthodes dans [7]. Notre approche, différente, est le sujet principal de notre thèse [5], soutenue à l'université de Paris VII sous la direction de A. Chenciner. Nous étudions la séparation des séparatrices d'un billard elliptique pour une déformation algébrique et donc globale de l'ellipse. Nous mesurons cette séparation par une fonction de Melnikov (dont le calcul est peu fréquent dans le cadre des systèmes dynamiques discrets), et nous observons que cette fonction est en général (lorsqu'elle admet une extension méromorphe) elliptique. Nous donnons ensuite sa décomposition en éléments simples avec les fonctions thêta. Nous obtenons ainsi sur un exemple tous les zéros d'une fonction de Melnikov, et donc toutes les orbites héterocliniques primaires.

We work with symplectic diffeomorphisms of the $n$-annulus ${\Bbb{A}}^n=T^*({\Bbb{R}}^n/{\Bbb{Z}}^n)$. Using the variational approach of Aubry and Mather, we are able to give a local description of the stable (and unstable) manifold for a hyperbolic fixed point. We use this in order to get a Melnikov-like formula for exact symplectic twist maps. This formula involves an infinite series that could be computed in some specific cases. We apply our formula to prove the existence of heteroclinic orbits for a family of twist maps in ${\Bbb{R}}^4$.

In dynamical systems examples are common in which two or more attractors coexist, and in such cases, the basin boundary is nonempty. When there are three basins of attraction, is it possible that every boundary point of one basin is on the boundary of the two remaining basins? Is it possible that all three boundaries of these basins coincide? When this last situation occurs the boundaries have a complicated structure. This phenomenon does occur naturally in simple dynamical systems. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. We introduce the basic notion of a ‘basin cell’. A basin cell is a trapping region generated by some well chosen periodic orbit and determines the structure of the corresponding basin. This new notion will play a fundamental role in our main results. We consider diffeomorphisms of a two-dimensional smooth manifold $M$ without boundary, which has at least three basins. A point $x\in M$ is a Wada point if every open neighborhood of $x$ has a nonempty intersection with at least three different basins. We call a basin $B$ a Wada basin if every $x\in\partial\bar{B}$ is a Wada point. Assuming $B$ is the basin of a basin cell (generated by a periodic orbit $P$), we show that $B$ is a Wada basin if the unstable manifold of $P$ intersects at least three basins. This result implies conditions for basins $B_{1},B_{2},\ldots,B_{N}(N\ge 3)$ to satisfy $\partial\bar{B}_{1}=\partial\bar{B}_{2}=\cdots =\partial\bar{B}_{N}$.

We consider $SL(2, {\Bbb R} )$-valued cocycles over rotations of the circle and prove that they are likely to have Lyapunov exponents $\approx \pm \log \lambda $ if the norms of all of the matrices are $\approx \lambda $. This is proved for $\lambda $ sufficiently large. The ubiquity of elliptic behavior is also observed.