We construct a minimal foliation of thin type which isnot uniquely ergodic. The notion of thin type relates to Rips'classification of foliations on 2-complexes.

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

The defect of a factor map out of a totally disconnected dynamical system isa number in$[0,\infty]$ which gives a quantitative measure of how well the dynamicalbehaviour of the factoragrees with that of the extension. The larger the defect, the more theydiffer. In this paper we obtainthe basic general properties of the defect and calculate it in a series ofspecific cases.

In this paper we show that the geodesic flow in a Hedlund-typemetric on the 3-torus possesses the shadowing property. This implies, inparticular, that any rotation vector is represented by a geodesic, afact that in the two-dimensional case is given by the Aubry–Mathertheory, while in the higher-dimensional case is still unknown.

We examine the question whether the dimension $D$ of a setor probability measure is the same as the dimension of itsimage under a typical smooth function, if the range spaceis at least $D$-dimensional. If $\mu$ is a Borelprobability measure of bounded support in ${\Bbb R}^n$with correlation dimension $D$, and if $m\geq D$, thenunder almost every continuously differentiable function(‘almost every’ in the sense of prevalence) from ${\BbbR}^n$ to ${\Bbb R}^m$, the correlation dimension of theimage of $\mu$ is also $D$. If $\mu$ is the invariantmeasure of a dynamical system, the same is true for almostevery delay coordinate map. That is, if $m\geq D$, then$m$ time delays are sufficient to find the correlationdimension using a typical measurement function. Further,it is shown that finite impulse response (FIR) filters donot change the correlation dimension. Analogous theoremshold for Hausdorff, pointwise, and information dimensions.We show by example that the conclusion fails forbox-counting dimension.

Nous développons dans cet article des résultats de M. B. Tabanov et moi-même annoncés dansla 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 VIIsous la direction de A. Chenciner.Nous étudions la séparation des séparatrices d'un billard elliptiquepour une déformation algébrique et donc globale de l'ellipse. Nous mesurons cette séparation par une fonction deMelnikov (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. Nousdonnons ensuite sa décomposition en éléments simples avec les fonctions thêta. Nous obtenonsainsi sur un exemple tous les zéros d'une fonction de Melnikov, et donc toutes les orbiteshéterocliniques primaires.

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

In this paper we will follow the bibliography of Blanchard [2] andgivean 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 piecewisemonotonic map, where $X$ is a finite union of closedintervals. Define $R(T)=\bigcap_{n=0}^{\infty}\overline{T^{-n}X}$, and suppose that $(R(T),T)$ hasa unique maximal measure $\mu$. The influence ofsmall perturbations of $T$ on the maximal measure isinvestigated. If $(R(T),T)$ has positive topologicalentropy, and if a certain stability condition issatisfied, then every piecewise monotonic map$\tilde{T}$, which is contained in a sufficientlysmall neighbourhood of $T$, has a unique maximalmeasure $\tilde{\mu}$, and the map$\tilde{T}\mapsto\tilde{\mu}$ is continuousat $T$.

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

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.

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

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

In dynamical systems examples are common in which two or moreattractors coexist, and in such cases, the basin boundary isnonempty. When there are three basins of attraction, is itpossible that every boundary point of one basin is on the boundaryof the two remaining basins? Is it possible that all threeboundaries of these basins coincide? When this last situationoccurs the boundaries have a complicated structure. Thisphenomenon does occur naturally in simple dynamical systems.The purpose of this paper is to describe the structure andproperties of basins and their boundaries for two-dimensionaldiffeomorphisms. We introduce the basic notion of a ‘basin cell’. Abasin cell is a trapping region generated by some well chosenperiodic orbit and determines the structure of the correspondingbasin. This new notion will play a fundamental role in our mainresults. We consider diffeomorphisms of a two-dimensionalsmooth manifold $M$ without boundary, which has at least threebasins. A point $x\in M$ is a Wada point if every openneighborhood of $x$ has a nonempty intersection with at leastthree different basins. We call a basin $B$ a Wada basin ifevery $x\in\partial\bar{B}$ is a Wada point. Assuming $B$ is thebasin of a basin cell (generated by a periodic orbit $P$), we showthat $B$ is a Wada basin if the unstable manifold of $P$intersects at least three basins. This result implies conditionsfor basins $B_{1},B_{2},\ldots,B_{N}(N\ge 3)$ to satisfy$\partial\bar{B}_{1}=\partial\bar{B}_{2}=\cdots=\partial\bar{B}_{N}$.

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 theaction bytranslations on the orbit closures of self-similar andself-affine tilingsof ${\Bbb R}^d$. The main focus is on spectral properties of suchsystemswhich are shown to be uniquely ergodic. We establishcriteria forweak mixing and pure discrete spectrum for wide classes ofsuch systems.They are applied to a number of examples which includetilings withpolygonal and fractal tile boundaries; systems with purediscrete,continuous and mixed spectrum.

We study the convergence to equilibrium states forcertain non-hyperbolic piecewise invertible systems.The multi-dimensional maps we shall considerdo not satisfy Renyi's condition (uniformlybounded distortion for any iterates) and do not necessarilysatisfy the Markov property.The failure of both conditions may cause singularities ofdensities of the invariant measures, even if they arefinite, and causes a crucial difficulty in applying thestandard technique of the Perron–Frobeniusoperator. Typical examples of maps we consideradmit indifferent periodic orbits and arise in many contexts.For the convergence of iterates ofthe Perron–Frobenius operator,we study continuity of the invariant density.

We study the Hausdorff dimension of invariantsets for expanding maps and that of hyperbolic sets on unstablemanifolds. Upper bounds for the Hausdorff dimension are given in termsof topological pressure, or topological entropy and Lyapunovexponents.

We consider $SL(2, {\Bbb R} )$-valued cocycles over rotations ofthe 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 ellipticbehavior is also observed.

Let $h$ be the Hausdorff dimension of the Julia set of a rational function$T$ with no non-periodic recurrentcritical 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 $m$. Themeasure $\mu$ is then proved to be ergodic and conservative and we study theset of those points whose all openneighborhoods haveinfinite measure $\mu$. Developing the concept of the inverse jumptransformation we show that the packing and Hausdorffdimensions of the conformal measure are equal to $h$. We also provide somesufficient conditions for Hausdorff and box dimensionsof the Julia set to be equal.