We construct a minimal foliation of thin type which is not uniquely ergodic. The notion of thin type relates to Rips' classification of foliations on 2-complexes.

The defect of a factor map out of a totally disconnected dynamical system is a number in $[0,\infty]$ which gives a quantitative measure of how well the dynamical behaviour of the factor agrees with that of the extension. The larger the defect, the more they differ. In this paper we obtain the basic general properties of the defect and calculate it in a series of specific cases.

A finite invariant set of a continuous map of an interval induces a permutation called its type. If this permutation is a cycle, it is called its orbit type. It has been shown by Geller and Tolosa that Misiurewicz–Nitecki orbit types of period $n$ congruent to $1$ (mod 4) and their generalizations to orbit types of period $n$ congruent to $3$ (mod 4) have maximal entropy among all orbit types of odd period $n$, and indeed among all permutations of period $n$. We further generalize this family to permutations of even period $n$ and show that they again attain maximal entropy amongst $n$-permutations.

In this paper we show that the geodesic flow in a Hedlund-type metric on the 3-torus possesses the shadowing property. This implies, in particular, that any rotation vector is represented by a geodesic, a fact that in the two-dimensional case is given by the Aubry–Mather theory, while in the higher-dimensional case is still unknown.

We examine the question whether the dimension $D$ of a set or probability measure is the same as the dimension of its image under a typical smooth function, if the range space is at least $D$-dimensional. If $\mu$ is a Borel probability measure of bounded support in ${\Bbb R}^n$ with correlation dimension $D$, and if $m\geq D$, then under almost every continuously differentiable function (‘almost every’ in the sense of prevalence) from ${\Bbb R}^n$ to ${\Bbb R}^m$, the correlation dimension of the image of $\mu$ is also $D$. If $\mu$ is the invariant measure of a dynamical system, the same is true for almost every delay coordinate map. That is, if $m\geq D$, then $m$ time delays are sufficient to find the correlation dimension using a typical measurement function. Further, it is shown that finite impulse response (FIR) filters do not change the correlation dimension. Analogous theorems hold for Hausdorff, pointwise, and information dimensions. We show by example that the conclusion fails for box-counting dimension.

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.

More than 30 years ago R\'enyi [1] introduced the representations of real numbers with an arbitrary base $\beta > 1$ as a generalization of the $p$-adic representations. One of the most studied problems in this field is the link between expansions to base $\beta$ and ergodic properties of the corresponding $\beta$-shift.

In this paper we will follow the bibliography of Blanchard [2] and give an affirmative answer to a question on the size of the set of real numbers $\beta$ having complicated symbolic dynamics of their $\beta$-shifts.

Let $T:X\to{\Bbb R}$ be a piecewise monotonic map, where $X$ is a finite union of closed intervals. Define $R(T)=\bigcap_{n=0}^{\infty} \overline{T^{-n}X}$, and suppose that $(R(T),T)$ has a unique maximal measure $\mu$. The influence of small perturbations of $T$ on the maximal measure is investigated. If $(R(T),T)$ has positive topological entropy, and if a certain stability condition is satisfied, then every piecewise monotonic map $\tilde{T}$, which is contained in a sufficiently small neighbourhood of $T$, has a unique maximal measure $\tilde{\mu}$, and the map $\tilde{T}\mapsto\tilde{\mu}$ is continuous at $T$.

We introduce and study a new class of dynamical systems, the projective billiards, associated with a smooth closed convex plane curve equipped with a smooth field of transverse directions. Projective billiards include the usual billiards along with the dual, or outer, billiards.

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 ${\Bbb{R}}^4$.

We consider the subset of the Julia set called the residual Julia set, which comes from an analogy of the residual limit set of a Kleinian group. We give a necessary and sufficient condition in order that the residual Julia set is empty in the case of hyperbolic rational functions.

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 one-dimensional flows which arise as hyperbolic invariant sets of a smooth flow on a manifold. Included in our data is the twisting in the local stable and unstable manifolds. A topological invariant sensitive to this twisting is obtained.

Let $M$ be a compact Riemannian manifold with no conjugate points such that its geodesic flow is expansive. We show that there exists a local product structure in the unit tangent bundle of the manifold which is invariant under the geodesic flow. In particular, we have that the set of closed geodesics is dense and that the flow is topologically transitive.

This paper investigates dynamical systems arising from the action by translations on the orbit closures of self-similar and self-affine tilings of ${\Bbb R}^d$. The main focus is on spectral properties of such systems which are shown to be uniquely ergodic. We establish criteria for weak mixing and pure discrete spectrum for wide classes of such systems. They are applied to a number of examples which include tilings with polygonal and fractal tile boundaries; systems with pure discrete, continuous and mixed spectrum.

We study the convergence to equilibrium states for certain non-hyperbolic piecewise invertible systems. The multi-dimensional maps we shall consider do not satisfy Renyi's condition (uniformly bounded distortion for any iterates) and do not necessarily satisfy the Markov property. The failure of both conditions may cause singularities of densities of the invariant measures, even if they are finite, and causes a crucial difficulty in applying the standard technique of the Perron–Frobenius operator. Typical examples of maps we consider admit indifferent periodic orbits and arise in many contexts. For the convergence of iterates of the Perron–Frobenius operator, we study continuity of the invariant density.

We study the Hausdorff dimension of invariant sets for expanding maps and that of hyperbolic sets on unstable manifolds. Upper bounds for the Hausdorff dimension are given in terms of topological pressure, or topological entropy and Lyapunov exponents.

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.

Let $h$ be the Hausdorff dimension of the Julia set of a rational function $T$ with no non-periodic recurrent critical points and let $m$ be the only $h$-conformal measure for $T$. We prove the existence of a $\sigma$-finite $T$-invariant measure $\mu$ equivalent with $m$. The measure $\mu$ is then proved to be ergodic and conservative and we study the set of those points whose all open neighborhoods have infinite measure $\mu$. Developing the concept of the inverse jump transformation we show that the packing and Hausdorff dimensions of the conformal measure are equal to $h$. We also provide some sufficient conditions for Hausdorff and box dimensions of the Julia set to be equal.