We use the methods developed with Lyubich for proving complex bounds for real quadratics to extend de Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for renormalizations of critical circle maps follows.

As another application of our methods we present a new proof of a theorem of Petersen on local connectivity of some Siegel Julia sets.

We describe a family of renormalization group transformations which apply to the problem of stability of invariant tori for analytic Hamiltonians. Each of these transformations corresponds to a fixed rotation vector, within a certain class of vectors in ${\mathbb R}^d$. As an application, we prove a KAM-type theorem that yields analytic invariant $d$-tori depending analytically on the Hamiltonian.

First, we show that there exists a sequence $(a_n)$ of integers which is a good averaging sequence in $L^2$ for the pointwise ergodic theorem and satisfies $$ \frac{a_{n+1}}{a_n}>e^{(\log n)^{-1-\epsilon}} $$ for $n>n(\epsilon)$. This should be contrasted with an earlier result of ours which says that if a sequence $(a_n)$ of integers (or real numbers) satisfies $$ \frac{a_{n+1}}{a_n}>e^{(\log n)^{-\frac{1}{2}+\epsilon}} $$ for some positive $\epsilon$, then it is a bad averaging sequence in $L^2$ for the pointwise ergodic theorem.

Another result of the paper says that if we select each integer $n$ with probability $1/n$ into a random sequence, then, with probability 1, the random sequence is a bad averaging sequence for the mean ergodic theorem. This result should be contrasted with Bourgain's result which says that if we select each integer $n$ with probability $\sigma_n$ into a random sequence, where the sequence $(\sigma_n)$ is decreasing and satisfies $$ \lim_{t\to\infty}\frac{\sum_{n\le t}\sigma_n}{\log t}=\infty, $$ then, with probability 1, the random sequence is a good averaging sequence for the mean ergodic theorem.

We study the behavior of multifractal spectra on the boundary of their domains of definition. In particular, we show that the dimension of the set of points having minimal (or maximal) pointwise dimension is not necessarily zero. However, this situation may be neglected in the sense of Baire category for Gibbs measures on expanding conformal repellers.

We introduce a set, $Q({\bf T})$, of Borel probability measures on the circle such that each $\mu\in Q({\bf T})$ obeys the conclusion of the Kerckhoff–Masur–Smillie theorem [3]: if $q$ is a meromorphic quadratic differential with at worst simple poles on a closed Riemann surface, then for each $\mu\in Q({\bf T})$ and $\mu$-a.e. $\zeta\in{\bf T}$, $\zeta q$ has uniquely ergodic vertical foliation. As an example, the normalized Cantor–Lebesgue measure belongs to $Q({\bf T})$. The analysis also yields an analogue, for the Teichmüller horocycle flow, of a theorem of Dani: every locally finite ergodic invariant measure for the Teichmüller horocycle flow is finite.

For hyperbolic systems and for Hölder cocycles with values in a compact metric group, we extend Livšic's periodic point characterisation of coboundaries. Here we show that two such cocycles are cohomologous when their respective ‘weights’ (of closed orbits) coincide. When it is only assumed that they are conjugate, one of the cocycles must (in general) be modified by an isomorphism (which stabilises conjugacy classes) to obtain cohomology. When the group is Lie and when a transitivity condition is satisfied, conjugacy of weights ensures that the cocycles are cohomologous with respect to a finitely extended group.

Let $(X,\phi)$ be a hyperbolic dynamical system and let $(G,\delta)$ be a Polish group. Motivated by Nicol and Pollicott, and then by Parry we study conditions under which two Hölder maps $f,g: X\longrightarrow G$ are Hölder cohomologous.

In the context of Nicol and Pollicott we show that if $f$ and $g$ are measurably cohomologous and the distortion of the metric $\delta $ by the cocycles defined by $f$ and $g$ is bounded in an appropriate sense, then $f$ and $g$ are Hölder cohomologous.

Two further results extend the main theorems recently presented by Parry. Under the hypothesis of bounded distortion we show that, if $f$ and $g$ give equal weight to all periodic points of $\phi $, then $f$ and $g$ are Hölder cohomologous. If the metric $\delta $ is bi-invariant, and if the skew-product $\phi _f$ defined by $f$ is topologically transitive, then conjugacy of weights implies that $g$ is Hölder conjugate to $\alpha \cdot f$ for some isometric automorphism $\alpha $ of $G$. The weaker condition that $g$-weights of periodic points are close to the identity whenever their $f$-weights are close to the identity implies that $g$ is continuously cohomologous to a homomorphic image of $f$.

The wreath-product construction is used to give a complete combinatorial description of the dynamics of period-doubling quadratic maps leading to the Feigenbaum map. An explicit description of the action on periodic points uses the Thue–Morse sequence. In particular, a wreath-product construction of this sequence is given. The combinatorial renormalization operator on the period-doubling family of maps is invertible.

Cocycles of $Z^m$ actions on compact metric spaces can be used to construct $R^m$ actions or flows, called suspension flows. A suspension provides a higher-dimensional analog to the familiar flow under a function and we look to this construction as a way of generating interesting $R^m$ flows. Even more importantly, an $R^m$ flow with a free dense orbit has an almost one-to-one extension which is a suspension [6] and thus suspensions can be used to model general $R^m$ flows. In this paper we examine the sensitivity of the suspension construction to small perturbations in the cocycle. Theorem 4.7 establishes the fact that two cocycles that are sufficiently close yield suspensions that are isomorphic up to a time change.

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

In [2], flows on the standard two-dimensional torus given by the differential equations \begin{equation*} \frac{dx}{dt}=a-Fy(x,y),\quad \frac{dv}{dt}=b+Fx(x,y) \end{equation*} were considered. It was assumed that $F(x,y)$ was real analytic and of period one in both $x$ and $y$. A key step in proving the results in [2] was to show that one could conclude topological transitivity for the flow provided one assumed: \begin{enumerate} \item[(a)] $a/b$ is irrational; \item[(b)] there does not exist a topological disc on the torus that is invariant under the flow. \end{enumerate}

We prove the following result. Let $G$ be a countable discrete group with finite conjugacy classes, and let $(X_n, n\in\mathbb Z)$ be a two-sided, strictly stationary sequence of $G$-valued random variables. Then $\mathscr T_\infty =\mathscr T_\infty ^*$, where $\mathscr T_\infty$ is the two-sided tail-sigma-field $\bigcap_{M\ge1}\sigma (X_m:|m|\ge M)$ of $(X_n)$ and $T_\infty ^*$ the tail-sigma-field $\bigcap_{M\ge0}\sigma (Y_{m,n}:m,n\ge M)$ of the random variables $(Y_{m,n}, m,n\ge0)$ defined as the products $Y_{m,n}=X_n\dots X_{-m}$. This statement generalises a number of results in the literature concerning tail triviality of two-sided random walks on certain discrete groups.

In this paper we show that the recent work of Dolgopiat and earlier work of the author give a rich class of examples of Axiom A attracting flows which mix exponentially fast. As an immediate corollary we have a proof of a conjecture of Ruelle.

Sur tout espace riemannien compact, localement symétrique de type non compact et de rang au moins égal à deux, nous construisons explicitement une métrique de Finsler donnant à l'espace le même volume que la métrique localement symétrique mais dont l'entropie volumique est strictement inférieure à l'entropie volumique de cette dernière. De plus, cette métrique de Finsler est l'unique minimum de l'entropie volumique parmi les métriques de Finsler $G$-invariantes normalisées par le volume de la variété.

D'autre part, concernant le rang un, nous prouvons que les métriques hyperboliques réelles sont des points critiques de l'entropie topologique parmi les métriques de Finsler sur une variété compacte, et ce, en normalisant aussi bien par le volume de la variété en dimension quelconque que par le volume de Liouville des fibrés unitaires en dimension deux.

Suppose that $\{\alpha_t\}_{t\in \mathbb{R}}$ is a flow on the compact metrizable space $X$. We prove that a necessary and sufficient condition for the existence of an embedding (injective $*$-homomorphism) of the crossed product $C(X)\rtimes_\alpha \mathbb{R}$ into some $AF$-algebra is that every point of $X$ be chain recurrent in the sense of Conley.

We present Sharkovskii's theorem for multidimensional perturbations of one-dimensional maps. We show that if an unperturbed one-dimensional map has a point of period $n$, then sufficiently close multidimensional perturbations of this map have periodic points of all periods which are allowed by Sharkovskii's theorem.

We prove that the system of two hard balls in a $\nu$-dimensional ($\nu\ge 2$) rectangular box is ergodic and, therefore, actually it is a Bernoulli flow.

Using pressure formulas we compute the Hausdorff dimension of the basic set of ‘almost every’ ${\cal C}^{1+\alpha}$ horseshoe map in $\mathbb{R}}^3$ of the form $F(x,y,z)= (\gamma(x,z), \tau(y,z), \psi(z))$, where $|\psi'|>1$ and $0< |\gamma'_x|, |\tau'_y| < \frac{1}{2}$ on the basic set. Similar results are obtained for attractors of nonlinear ‘baker's maps’ in $\mathbb{R}}^3$.

Lemma 2.2 on p. 704 was quoted from the Ph.D. thesis of B. Praggastis [2, 1.4] incorrectly. The local isomorphism property ($=$ repetitivity) of a $\phi$-subdividing tiling does imply that the subdivision matrix $M$ is primitive (that is, $M^k>0$ for some $k\ge1$), however, the converse is false, in general. I am grateful to A. E. Robinson Jr. who showed me a counter-example.

It is known that in generic, full unimodal families with a critical point of finite order, there exists a set of positive measure in parameter space such that the corresponding maps have chaotic behaviour. In this paper we prove the corresponding statement for certain families of unimodal maps with flat critical point. One of the key-points is a large deviation argument for sums of ‘almost’ independent random variables with only finitely many moments.