We study minimal homeomorphisms (all orbits are dense) of the tori Tn, n≤4. The linear part of a homeomorphism φ of Tn is the linear mapping L induced by φ on the first homology group of Tn. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of L if φ minimal. We show that if φ is minimal and n≤4, then L is quasi-unipontent, that is, all of the eigenvalues of L are roots of unity and conversely if L∈GL(n,ℤ) is quasi-unipotent and 1 is an eigenvalue of L, then there exists a C∞ minimal skew-product diffeomorphism φ of Tn whose linear part is precisely L. We do not know whether these results are true for n≥5. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation.

We show that dynamical systems with $\unicode[STIX]{x1D719}$-mixing measures have local escape rates which are exponential with rate 1 at non-periodic points and equal to the extremal index at periodic points. We apply this result to equilibrium states on subshifts of finite type, Gibbs–Markov systems, expanding interval maps, Gibbs states on conformal repellers, and more generally to Young towers, and by extension to all systems that can be modeled by a Young tower.

We generalize the notion of reversible systems in symbolic dynamics, and investigate their properties. It is shown that a reversible topological Markov shift can be represented by a pair of matrices of special types. This enables us to classify the invariant measures of the reversible systems. Necessary and/or sufficient conditions for the existence of a reversal of finite order are established in terms of the adjacency matrices. We also prove that a topological Markov shift with a reversal of order two admits reversals of all orders.

We consider a new orbit equivalence invariant for measure-preserving actions of groups on the probability space, $\sigma:G\rightarrow {\rm Aut}(X,\mu)$, denoted by $\chi_0(\sigma;G)$ and defined as the ‘intersection’ of the 1-cohomology group, H$^1(\sigma,G)$, with Connes' invariant, $\chi(M)$, of the cross product von Neumann algebra, $M=L^\infty(X,\mu)\rtimes_\sigma G$. We calculate $\chi_0(\sigma;G)$ for certain actions of groups of the form $G=H\times K$ with $H$ non-amenable and $K$ infinite amenable and we deduce that any such group has uncountably many orbit inequivalent actions.

Let ${\cal R}$ be an ergodic discrete equivalence relation on a Lebesgue space and $\alpha$ its cocycle with values in a locally compact group $G$. We say that an automorphism of ${\cal R}$ is compatible with $\alpha$ if it preserves the cohomology class of $\alpha$. We introduce a quasi-order relation on the set of all cocycles of ${\cal R}$ by means of comparison of the corresponding groups of all automorphisms being compatible with them. We find simple necessary and sufficient conditions under which two cocycles of a hyperfinite measure-preserving equivalence relation with values in compact (possibly different) groups are connected by this relation. Next, given an ergodic subrelation ${\cal S}$ of ${\cal R}$, we investigate the problem of extending ${\cal S}$-cocycles up to ${\cal R}$-cocycles and improve the recent results of Gabriel–Lemańczyk–Schmidt. As an application we study the problem of lifting of automorphisms of ${\cal R}$ up to automorphisms of the skew product ${\cal R}\times_\alpha G$, provide new short proofs of some known results and answer several questions from [ALV, ALMN].

We answer a question proposed by Barge and Franks in 1993 by showing that a bounded connected set in the plane which is closed or open and meets an infinite number of its iterates under an orientation-preserving fixed point free homeomorphism must meet all of them.

Let $T$ be an $\mathbb{R}$-tree with dense orbits in the boundary of outer space. When the free group $\mathbb{F}_{N}$ acts freely on $T$, we prove that the number of projective classes of ergodic currents dual to $T$ is bounded above by $3N-5$. We combine Rips induction and splitting induction to define unfolding induction for such an $\mathbb{R}$-tree $T$. Given a current ${\it\mu}$ dual to $T$, the unfolding induction produces a sequence of approximations converging towards ${\it\mu}$. We also give a unique ergodicity criterion.

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.

We prove that if $F$ is an orientation-preserving homeomorphism of the plane that leaves invariant an irreducible plane separating continuum $\Delta$, then, with the possible exception of three numbers, if $p/q$ is a reduced rational in the interior of the convex hull of the rotation set of $F\vert_{\Delta}$ (with respect to some lift) there are at least two distinct periodic orbits of $F\vert_{\Delta}$ of period $q$ and rotation number $p/q$. This result also applies to certain nonseparating invariant continua.

Given a lamination in a compact space and a laminated vector field $X$ which is hyperbolic when restricted to the leaves of the lamination, we distinguish a class of $X$-invariant probabilities that describe the behavior of almost every $X$-orbit in every leaf, which we call Gibbs $u$-states. We apply this to the case of foliations in compact manifolds having leaves with negative curvature, using the foliated hyperbolic vector field on the unit tangent bundle to the foliation generating the leaf geodesics. When the Lyapunov exponents of such ergodic Gibbs $u$-states are negative, it is an SRB measure (having a positive Lebesgue basin of attraction). When the foliation is by hyperbolic leaves, this class of probabilities coincide with the classical harmonic measures introduced by Garnett. Furthermore, if the foliation is transversally conformal and does not admit a transverse invariant measure we show that there are finitely many ergodic Gibbs $u$-states, each supported in one minimal set of the foliation, each having negative Lyapunov exponents, and the union of their basins of attraction has full Lebesgue measure. The leaf geodesics emanating from a point have a proportion whose asymptotic statistics are described by each of these ergodic Gibbs $u$-states, giving rise to continuous visibility functions of the attractors. Reversing time, by considering $-X$, we obtain the existence of the same number of repellers of the foliated geodesic flow having the same harmonic measures as projections to $M$. In the case of only one attractor, we obtain a north to south pole dynamics.

Inspired by topological Wiener–Wintner theorems we study the mean ergodicity of amenable semigroups of Markov operators on $C(K)$ and show the connection to the convergence of strong and weak ergodic nets. The results are then used to characterize mean ergodicity of Koopman semigroups corresponding to skew product actions on compact group extensions.

In this paper, we consider tilings of the hyperbolic 2-space ${\mathbb H}^2$, built with a finite number of polygonal tiles, up to affine transformation. To such a tiling $T$, we associate a space of tilings: the continuous hull $\Omega(T)$ on which the affine group acts. This space $\Omega(T)$ inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First, we prove that the finite harmonic measures of this laminated space correspond to finite invariant measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally, we give examples with an arbitrary number of ergodic invariant probability measures.

We study ergodic finite and infinite measures defined on the path space $X_{B}$ of a Bratteli diagram $B$ which are invariant with respect to the tail equivalence relation on $X_{B}$. Our interest is focused on measures supported by vertex and edge subdiagrams of $B$. We give several criteria when a finite invariant measure defined on the path space of a subdiagram of $B$ extends to a finite invariant measure on $B$. Given a finite ergodic measure on a Bratteli diagram $B$ and a subdiagram $B^{\prime }$ of $B$, we find the necessary and sufficient conditions under which the measure of the path space $X_{B^{\prime }}$ of $B^{\prime }$ is positive. For a class of Bratteli diagrams of finite rank, we determine when they have maximal possible number of ergodic invariant measures. The case of diagrams of rank two is completely studied. We also include an example which explicitly illustrates the proven results.

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 establish several new characterizations of amenable $W^*$ - and $C^*$ -dynamical systems over arbitrary locally compact groups. In the $W^*$ -setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of $(M,G,\alpha )$ converging point weak* to the identity of $G\bar {\ltimes }M$ . In the $C^*$ -setting, we prove that amenability of $(A,G,\alpha )$ is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$ , as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted $C^*$ -dynamical systems, Hilbert $C^*$ -modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When $Z(A^{**})=Z(A)^{**}$ , it follows that amenability is equivalent to the 1-positive approximation property of Exel and Ng [Approximation property of $C^*$ -algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,\alpha )$ coincides with topological amenability of the G-space $(G,X)$ .

For a transcendental entire function $f$ of finite order in the Eremenko–Lyubich class ${\mathcal{B}}$, we give conditions under which the Lebesgue measure of the escaping set ${\mathcal{I}}(f)$ of $f$ is zero. This complements the recent work of Aspenberg and Bergweiler [Math. Ann. 352(1) (2012), 27–54], in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko–Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to certain infinite-order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.

Every sufficiently regular non-periodic space of tilings of $\mathbb {R}^d$ has at least one pair of distinct tilings that are asymptotic under translation in all the directions of some open $(d-1)$-dimensional hemisphere. If the tiling space comes from a substitution, there is a way of defining a location on such tilings at which asymptoticity ‘starts’. This leads to the definition of the branch locus of the tiling space: this is a subspace of the tiling space, of dimension at most $d-1$, that summarizes the ‘asymptotic in at least a half-space’ behavior in the tiling space. We prove that if a $d$-dimensional self-similar substitution tiling space has a pair of distinct tilings that are asymptotic in a set of directions that contains a closed $(d-1)$-hemisphere in its interior, then the branch locus is a topological invariant of the tiling space. If the tiling space is a two-dimensional self-similar Pisot substitution tiling space, the branch locus has a description as an inverse limit of an expanding Markov map on a zero- or one-dimensional simplicial complex.

We consider a C2 non-renormalizable polymodal map with finitely many non-flat critical points of turning type and we prove that any minimal set of f has zero Lebesgue measure.

In this article, we prove a preparation theorem for functions which admit a certain type of expansion called Chebychev expansion. Taylor expansions are particular cases of Chebychev expansions. The result is based on an approach essentially different from those used for the classical preparation theorems. It has applications in bifurcation theory of vector fields.

Generalizing from the centralizer of a measure-preserving dynamical system (X, ℬ, μ, T), one defines the Kakutani centralizer KC(T) of all ‘even Kakutani factor maps’ ϕ from T to itself. Such a ϕ is a composition φ2ϕ1 of an even Kakutani orbit equivalence ϕ1 and a factor map ϕ2. We construct here an ergodic T acting on a nonatomic Lebesgue space (X,ℬ,μ) with the property that any ϕ ∈ KC(T) is invertible and of the form

All invertible maps of this form are automatically in KC(T) and hence for this T the Kakutani centralizer is as small as possible.