We study automorphisms of the CAR-algebra obtained from binary shifts. We consider cases when the $C^\ast$-dynamical system is asymptotically abelian, is proximally asymptotically abelian, is an entropic $K$-system or has completely positive entropy. The entropy is computed in several cases.

For a large class of Markov operators on trees we prove the formula ${\bf HD}\,\nu=h/l$ connecting the Hausdorff dimension of the harmonic measure $\nu$ on the tree boundary, the rate of escape $l$ and the asymptotic entropy $h$. Applications of this formula include random walks on free groups, conditional random walks, random walks in random environment and random walks on treed equivalence relations.

Let $F$ be a nonquasi-unipotent one-parameter (cyclic) subgroup of a unimodular Lie group $G$, $\Gamma$ a discrete subgroup of $G$. We prove that for certain classes of subsets $Z$ of the homogeneous space $G/\Gamma$, the set of points in $G/\Gamma$ with $F$-orbits staying away from $Z$ has full Hausdorff dimension. From this we derive applications to geodesic flows on manifolds of constant negative curvature.

Recently, Naito considered quasi-periodic orbits of Hölder continuous functions and obtained results relating the exponent in the Hölder condition to the asymptotic behaviour of the inclusion lengths of the $\epsilon$-almost periods of these orbits, and also to the fractal dimension of these orbits. In this paper we improve these results.

In this paper we obtain results about the regularity of the transfer map between two cocycles over an Anosov system, with values in either a diffeomorphism or a Lie group. We also explain how certain examples of de la Llave show that our results are essentially optimal.

In 1984, Gromov (see [4] and [6]) introduced the idea of subdividing a ‘branching’ polyhedron into smaller cells and replacing these cells by more complex objects, reminiscent of the growth of multicellular organisms in biology. The simplest situation of this kind is a graph substitution which replaces certain subgraphs in a graph $G$ by bigger finite graphs. The most basic graph substitution is a vertex replacement rule which replaces certain vertices of $G$ with finite graphs. This paper develops a framework for studying vertex replacements and discusses the asymptotic behavior of iterated vertex replacements, the limit objects, and the induced dynamics on the space of infinite graphs from the viewpoint of geometry and dynamical systems.

A sequence of random variables, $Y_0,Y_1,Y_2,\ldots\,$, is called standard if there exists a one-sided isomorphism between it and a sequence of independent random variables. In this paper it is demonstrated that the sequence arising from the past of the $T,T^{-1}$ map is not standard.

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 study spectral properties of the flow $\dot x =1/F(x,y)$, $\dot y = 1/\lambda F(x,y)$ on the 2-torus. We show that, in general, the speed of approximation in cyclic approximation gives an upper bound on the Hausdorff dimension of the supports of spectral measures. We use this to prove that for generic pairs $(F,\lambda)$ the spectrum of the flow on the torus is singular continuous with all spectral measures supported on sets of zero Hausdorff dimension.

We prove that certain topologically mixing two-dimensional shifts of finite type have a ‘fundamental’ $1$-cocycle with the property that every continuous $1$-cocycle on the shift space with values in a discrete group is continuously cohomologous to a homomorphic image of the fundamental cocycle. These fundamental cocycles are closely connected with representations of the shift space by Wang tilings and the tiling groups of Conway, Lagarias and Thurston, and they determine the projective fundamental groups of the shift spaces introduced by Geller and Propp.

Let $F_\lambda(z)= \lambda z + {\cal O}(z^2)$ be a one parameter holomorphic family of holomorphic maps defined in a neighborhood of $\lambda_0* \overline{\mbox{D}}$. Assume that $F_{\lambda_0}$ has a Siegel disc $\Delta_0$, then the orbit of a point in the interior of the Siegel disc can be followed very closely by periodic orbits of nearby maps. The same technique is applied to Herman rings and Cremer points.

Let $\rho_0$ be the standard action of a higher-rank lattice $\Gamma$ on a torus by automorphisms induced by a homomorphism $\pi_0:\Gamma\to SL(n,{\Bbb Z})$. Assume that there exists an abelian group ${\cal A}\subset \Gamma$ such that $\pi_0({\cal A})$ satisfies the following conditions: (1) ${\cal A}$ is ${\Bbb R}$-diagonalizable; (2) there exists an element $a\in {\cal A}$, such that none of its eigenvalues $\lambda_1,\dots,\lambda_n$ has unit absolute value, and for all $i,j,k=1,\dots,n$, $|\lambda_i\lambda_j|\neq|\lambda_k|$; (3) for each Lyapunov functional $\chi_i$, there exist finitely many elements $a_j\in {\cal A}$ such that $E_{\chi_i}=\cap_{j} E^u(a_j)$ (see \S1 for definitions). Then $\rho_0$ is locally rigid. This local rigidity result differs from earlier ones in that it does not require a certain one-dimensionality condition.

We study ${\Bbb Z}^2$ actions arising as base point actions of tiling representations of ${\Bbb R}^2$ flows. We cast an equivalence relation between such actions in terms of a simple arithmetic condition on an orbit equivalence. Stated as such, our equivalence class is easily seen to be a restricted even Kakutani equivalence, as we well as a higher-dimensional generalization of $\alpha$-equivalence, defined by Fieldsteel, del Junco and Rudolph for ${\Bbb Z}$ actions.

We use the concept of enveloping Ellis semigroups in order to classify the behaviour of dynamical systems on intervals. Several notions of chaos and non-chaos are expressed by means of algebraic properties of the corresponding Ellis semigroups.

The Collet–Eckmann condition is quasisymmetrically invariant: if an S-unimodal map $f$ is quasisymmetrically conjugate to an S-unimodal map $g$ then $f$ satisfies the Collet–Eckmann condition if and only if $g$ satisfies the Collet–Eckmann condition.

In this paper we consider a smooth dynamical system $f$ and give estimates of the growth rates of vector fields and differential forms in the $L_p$ norm under the action of the dynamical system in terms of entropy, topological pressure and Lyapunov exponents. We prove a formula for the topological entropy $$h_{\rm top}=\lim_{n\to\infty} \frac 1n \log \int \Vert Df_x^n\,^{\wedge}\Vert \,dx,$$ where $Df_x^n\,^{\wedge}$ is a mapping between full exterior algebras of the tangent spaces. An analogous formula is given for the topological pressure.

We introduce and study a class of endomorphisms which are piecewise smooth and have hyperbolic attractors. We prove the existence of SBR measures and develop the stable manifold theory and ergodic theory of such maps. We show that the Young dimension formula holds if and only if the map is invertible SBR-almost everywhere.

We characterize the set of $n$-jets admitting an extension which is a germ of a differential equation with an analytic first integral, and compute its codimension in the $n$-jet space. Some applications in the case of the centre-focus problem are given.

We define 2-critically finite maps of ${\bf P}^2$ and show that they have no nontrivial closed backward invariant sets. In particular, their Julia sets $J_1$, defined as the support of a natural invariant measure, are equal to ${\bf P}^2$. We also prove that repelling periodic points are dense for such maps.

In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise ergodic theorem and give additional information such as control of the number of upcrossings of the ergodic averages. Related results for differentiation and for the connection between differentiation operators and the dyadic martingale are also established.