In this paper, we study the behaviour of the entropy function of higher-dimensional shifts of finite type. We construct a topologically mixing $\mathbb{Z}^2$ shift of finite type whose ergodic invariant measures are connected in the $\overline{d}$ topology and whose entropy function has a strictly local maximum. We also construct a topologically mixing $\mathbb{Z}^2$ shift of finite type X with the property that there is a uniform gap between the topological entropy of X and the topological entropy of any subshift of X with stronger mixing properties. Our examples illustrate the necessity of strong topological mixing hypotheses in existing higher-dimensional representation and embedding theorems.

A dimension invariant for isomorphism of non-singular ergodic integer actions is presented. A result analogous to Shannon's Theorem is used to calculate the critical dimension of type III odometer actions. It is shown that the dimension is non-trivial and that it is not an invariant of orbit equivalence.

Soient G un groupe de Lie semi-simple de centre fini et \Gamma un sous-groupe discret Zariski dense de G. Dans des travaux précédents, nous avons défini l'indicateur de croissance de \Gamma, objet qui généralise en rang supérieur l'exposant critique considéré en \mathbb{R}-rang 1, et nous avons associé à \Gamma des généralisations des mesures de Patterson-Sullivan. Dans cet article, nous donnons des précisions sur ces objets quand \Gamma est un sous-groupe du type de Schottky, en les reliant au formalisme thermodynamique.

For diffeomorphisms on surfaces with basic sets, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point in the basic set then the conjugacy has a smooth extension to the surface. These results generalize the similar ones of D. Sullivan, E. de Faria and ours for one-dimensional expanding dynamics.

Assuming that two ergodic automorphisms T and R are disjoint, we prove that an ergodic extension $\tilde{T}$ of T remains disjoint with R when, in a representation of $\tilde{T}$ as a skew product over T, the fiber automorphisms belong to a mildly mixing action of a second countable locally countable abelian (LCA) group and the corresponding cocycle with values in this group is ergodic. This allows us to show that the class of ergodic automorphisms which are disjoint from all weakly mixing automorphisms is not closed under taking ergodic joinings, thereby solving a problem stated by E. Glasner and B. Weiss.

Let $\Gamma$ be a lattice in $G={\rm SL}(n,\mathbb{R})$ and X = G/S be a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish the uniform distribution of orbits of $\Gamma$ in X analogous to the classical equidistribution on a torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of a Borel subgroup of G.

A locally free Lie group action on a closed manifold is called parameter rigid, if given another action with the same orbit foliation, there is a $C^\infty$ orbit-preserving diffeomorphism which conjugates one action to the other, up to an automorphism of the Lie group. We show some relationships between parameter rigidity and the first leafwise cohomology of the orbit foliation, and give a new example of a parameter rigid solvable group action. We also compute the first leafwise cohomology of the stable foliation of geodesic flows of closed orientable surfaces of constant negative curvature.

We prove that the solenoid with two different contraction coefficients has zero Hausdorff and positive packing measure in its own dimension and the SBR measure is equivalent to the packing measure on the attractor. Further, we prove similar statements for Slanting Baker maps with intersecting cylinders (in \mathbb{R}^{2}).

On montre que la condition combinatoire de fortes coïncidences est suffisante pour qu'un système substitutif de type Pisot soit isomorphe en mesure à un échange de morceaux dans un compact autosimilaire de mesure non nulle dans le produit d'un espace euclidien et d'extensions finies de corps p-adiques. En particulier, tout système substitutif de type Pisot est une extension finie de son facteur équicontinu maximal ; en règle générale, ce dernier contient une translation p-adique si et seulement si la matrice d'incidence de la substitution est nilpotente modulo p.

We construct d-dimensional substitution sequences for which the continuous part of the spectrum is generated by measures equal to Lebesgue measure. A special case is the Rudin–Shapiro substitution sequence. The construction uses Hadamard matrices in an essential way, so the dimension and size of a substitution is restricted by the size of the Hadamard matrix defining it. Each such substitution automatically has a dual substitution, which is defined by the same Hadamard matrix, and which retains a Lebesgue spectral component. We also see that random application of our substitutions produces sequences with a Lebesgue component in their spectrum. Finally, we see that any d-dimensional substitution with d > 1 can be ‘unraveled’ into lower-dimensional substitutions which still have Lebesgue spectral components.

A symbolic matrix system and a $\lambda$-graph system for a subshift are generalizations of a symbolic matrix and a $\lambda$-graph (= finite symbolic matrix); for a sofic shift respectively. In [K. Matsumoto. (Presentations of subshifts and their topological conjugacy invariants. Doc. Math. 4 (1999), 285–340)], two equivalence relations between symbolic matrix systems were formulated, properly strong shift equivalence and strong shift equivalence, which are related to topological conjugacy of subshifts. In this paper, we first prove that these two equivalence relations are equivalent to each other for a large class of symbolic matrix systems. State splitting of a $\lambda$-graph system is introduced as a generalization of state splitting of a finite directed graph. State splitting gives rise to a strong shift equivalence for the corresponding symbolic matrix systems. We prove that two subshifts are topologically conjugate if and only if the canonical $\lambda$-graph system for one subshift is obtained by a finite sequence of merged state splittings and merged state amalgamations from the canonical $\lambda$-graph system for the other subshift.

In dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is non-empty. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well-chosen periodic orbit), then the basin consists of a central body (the basin cell) and a finite number of channels attached to it and the basin boundary is fractal. We prove the following surprising property for certain fractal basin boundaries: a basin of attraction B has a basin cell if and only if every diverging path in basin B has the entire basin boundary \partial\bar{B} as its limit set. The latter property reflects a complete entangled basin and its boundary.

We prove that any finitely generated solvable group of zero entropy contains a nilpotent subgroup of finite index. In particular, any finitely generated solvable group of exponential growth is of uniformly exponential growth.

Let $r \geq 1$; the Følner sequences $\{F^{n,1}\}_{n=1}^{\infty}, \{F^{n,2}\}_{n=1}^{\infty}, \dotsc, \{F^{n,r}\}_{n=1}^{\infty}$ satisfy the bounded intersection property if there is a constant p such that, for any $n\in\mathbb{N}$ and $1\leq i\leq r$, each $F^{n,i}$ can intersect no more than p disjoint translates of $F^{n,1}, F^{n,2},\dotsc, F^{n,r}$. They have comparable magnitudes if $0<\varliminf_{n}|F^{n,i}|/|F^{n,j}|<\infty$ for $1\leq i,j\leq r$. Suppose that G is a countable Abelian group with an element of infinite order, and let $\mathcal{X}$ be a mixing finite (or $\beta$-local, with $\beta>1/2$) rank action of G on a probability space. Suppose that the Følner sequences of $\mathcal{X}$ satisfy the bounded intersection property, and have comparable magnitudes. Then $\mathcal{X}$ is mixing of all orders. We follow Ryzhikov's joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of k copies of $\mathcal{X}$ is necessarily product measure.

We prove measurable Livšic theorems for Lie group valued cocycles over certain classes of uniformly hyperbolic maps with discontinuities, in particular the $\beta$-transformation, C2 Markov maps and a simple class of toral linked twist maps. As an application, we establish stable ergodicity results for compact group extensions of certain classes of uniformly hyperbolic systems with discontinuities.

We investigate properties of a class of martingales formed by picking a measurable set A in a compact group G, taking random rotates of A, and taking measures of the resulting intersections, suitably normalized. Such martingales are shown to yield measures on G^\infty that are singular with respect to product Haar measure, and their ergodic decompositions with respect to the shift map are analyzed. Connections are made with De Finetti theory.

We prove that a generic C1 vector field on a closed 3-manifold either has infinitely many sinks or sources or else is singular Axiom A without cycles. Singular Axiom A means that the non-wandering set of the vector field has a decomposition into compact invariant sets, each being either a hyperbolic basic set or a singular hyperbolic attractor (like the Lorenz-like ones) or a singular hyperbolic repeller. An attractor is a transitive set which attracts all nearby future orbits, and a repeller is an attractor for the time-reversed flow. Our result implies that generic C1 vector fields on closed 3-manifolds do exhibit either attractors or repellers.

We show that any Pisot substitution dynamical system on two symbols has pure discrete spectrum. This follows from an analysis of a combinatorial ‘balanced pair algorithm’, in view of recent progress on the Strong Coincidence Conjecture made by Barge and Diamond.

We generalize the average asymptotic linking number of a pair of divergence-free vector fields on homology three-spheres by considering the linking of a divergence-free vector field on a manifold of arbitrary dimension with a codimension-two foliation endowed with an invariant transverse measure. We prove that the average asymptotic linking number is given by an integral of Hopf type. Considering appropriate vector fields and measured foliations, we obtain an ergodic interpretation of the Godbillon–Vey invariant of a family of codimension-one foliations discussed by Kotschick (Symplectic and Contact Topology: Interactions and Perspectives. Eds. Y. Eliashberg, B. Khesin and F. Lalonde. American Mathematical Society, Providence, RI).

We prove that fairly general spaces of tilings of \mathbb{R}^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in the second author's recent work, and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a \mathbb{Z}^d subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that (1) the tiles are assumed to be polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to full-face), (2) only a finite number of tile types are allowed, and (3) each tile type appears in only a finite number of orientations. The proof is constructive and we illustrate it by constructing a ‘square’ version of the Penrose tiling system.