We consider transitive Anosov flows φ: M → M and give necessary and sufficient conditions for every homology class in H1(M,ℤ) to contain a closed φ-orbit. Under these conditions, we derive an asymptotic formula for the number of closed φ-orbits in a fixed homology class, generalizing a result of Katsuda and Sunada.

In the study of a dynamical system f: X → X generated by a continuous map f on a compact metric space X, the chain recurrent set is an object of fundamental interest. This set was defined by C. Conley, who showed that it has two rather different looking, but equivalent, definitions: one given in terms of ‘approximate orbits’ through individual points (pseudo-orbits, or ε-chains), and the other given in terms of the global structure of the class of ‘attractors’ and ‘basins of attraction’ of f. The first of these definitions generalizes directly to dynamical systems on any metric space, compact or not. The main purpose of this paper is to extend the second definition to non-compact spaces in such a way that it remains equivalent to the first.

This paper explores the effect of various graphical constructions upon the associated graph C*-algebras. The graphical constructions in question arise naturally in the study of flow equivalence for topological Markov chains. We prove that out-splittings give rise to isomorphic graph algebras, and in-splittings give rise to strongly Morita equivalent C*-algebras. We generalize the notion of a delay as defined in (D. Drinen, Preprint, Dartmouth College, 2001) to form in-delays and out-delays. We prove that these constructions give rise to Morita equivalent graph C*-algebras. We provide examples which suggest that our results are the most general possible in the setting of the C*-algebras of arbitrary directed graphs.

For a homeomorphism of a compact metrizable space X, we show that the property that every point of X is pseudo-non-wandering (see definition 2) is equivalent to the possibility of embedding the corresponding transformation group C*-algebra into an AF-algebra.

We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we give an explicit description of all ergodic probability measures that are invariant with respect to the tail equivalence relation (or the Vershik map); these measures are completely described by the incidence matrix of the Bratteli diagram. Since such diagrams correspond to substitution dynamical systems, our description provides an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

Kupka-Smale like theorems are proven in the real analytic case, using existing perturbation schemes for the smooth case and the heat operator. As a consequence, a topological proof is obtained of Siegel's theorem on the generic divergence of normal form transformations.

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma$ by automorphisms of compact abelian groups in terms of Fuglede–Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma$-action by means of a ‘fundamental homoclinic point’ and the description of entropy in terms of the renormalized logarithmic growth rate of the set of $\Gamma_n$-fixed points, where $(\Gamma_n,n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma$ with trivial intersection.

We prove that the crossed product of the CAR algebra M2∞ by the shift is an inductive limit of homogeneous algebras over the circle with fibres full matrix algebras. As a consequence the crossed product has real rank zero, and where is the Cuntz algebra of order 2.

In previous work, the author introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here, it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit change σ-algebra.

We prove a non-conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.

We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterize continuous orbit equivalence in terms of isomorphisms of $C^{\ast }$-crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Second, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this is not always the case by constructing topological dynamical systems (actions of free abelian groups and also non-abelian free groups) that are continuously orbit equivalent but not conjugate. Furthermore, we prove positive rigidity results. For instance, for solvable duality groups, general topological Bernoulli actions and certain subshifts of full shifts over finite alphabets are rigid.

We show that a smooth compact Riemannian manifold of dimension greater than or equal to 2 admits a Bernoulli diffeomorphism with non-zero Lyapunov exponents.

We study the problem of conservation of maximal and lower-dimensional invariant tori for analytic convex quasi-integrable Hamiltonian systems. In the absence of perturbation the lower-dimensional tori are degenerate, in the sense that the normal frequencies vanish, so that the tori are neither elliptic nor hyperbolic. We show that if the perturbation parameter is small enough, for a large measure subset of any resonant submanifold of the action variable space, under some generic non-degeneracy conditions on the perturbation function, there are lower-dimensional tori which are conserved. They are characterized by rotation vectors satisfying some generalized Bryuno conditions involving also the normal frequencies. We also show that, again under some generic assumptions on the perturbation, any torus with fixed rotation vector satisfying the Bryuno condition is conserved for most values of the perturbation parameter in an interval small enough around the origin. According to the sign of the normal frequencies and of the perturbation parameter the torus becomes either hyperbolic or elliptic or of mixed type.

The Bowen-Margulis measure on the unit tangent bundle of the universal covering of a compact manifold of negative curvature is determined by its restriction to the leaves of the strong unstable foliation. We describe this restriction to any strong unstable manifold W as a spherical measure with respect to a natural distance on W.

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.

Let Г be a finitely generated non-elementary Fuchsian group acting in the disk. With the exception of a small number of co-compact Г, we give a representation of g ∈ Г as a product of a fixed set of generators Гo in a unique shortest ‘admissible form’. Words in this form satisfy rules which after a suitable coding are of finite type. The space of infinite sequences Σ of generators satisfying the same rules is identified in a natural way with the limit set Λ of Г by a map which is bijective except at a countable number of points where it is two to one. We use the theory of Gibbs measures onΣ to construct the so-called Patterson measure on Λ [8], [9]. This measure is, in fact, Hausdorff 5-dimensional measure on Λ, where S is the exponent of convergence of Г.

We consider the crossed product or transformation group C*-algebras arising from actions of the group of integers on a totally disconnected compact metrizable space. Under a mild hypothesis, we give a necessary and sufficient dynamical condition for the invertibles in such a C*-algebra to be dense. We also examine the property of residual finiteness for such C*-algebras.

In this article, we study the automorphism group $\text{Aut}(X,{\it\sigma})$ of subshifts $(X,{\it\sigma})$ of low word complexity. In particular, we prove that $\text{Aut}(X,{\it\sigma})$ is virtually $\mathbb{Z}$ for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a $d$-step nilsystem is nilpotent of order $d$ and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually $\mathbb{Z}$.

This is a partial survey of various developments of the classical KAM theory in dynamical systems to infinite-dimensional lattice models and Hamiltonian PDEs. Particular emphasis is given on Nekhoroshev stability and growth of higher Sobolev norms. Some new results are also presented.

We find a class of C∞ maps of an interval with zero topological entropy and chaotic in the sense of Li and Yorke.