In this paper we compute the Hausdorff dimension of the set $D(\varphi_n)$ of points on divergent trajectories of the homogeneous flow $\varphi_n$ induced by the one-parameter subgroup $\mathop{\rm diag}(e^t,e^{-t})$ acting by left multiplication on the product space $G^n/\Gamma^n$, where $G=\mathop{\rm SL}(2,{\mathbb R})$ and $\Gamma=\mathop{\rm SL}(2,{\mathbb Z})$. We prove that $\dim_H D(\varphi_n)=3n-\frac{1}{2}$ for $n\ge2$.

The classical KAM theorem establishes the persistence of invariant Lagrangean tori in nearly integrable Hamiltonian systems. These tori are quasi-periodic with Diophantine frequency vectors and their union is a nowhere dense set of positive measure in phase space. A long-standing question is to what extent the perturbed tori are unique. Using the fact that at the level of tori there exists a Whitney smooth conjugacy between the integrable approximation and its perturbation, we are able to prove this unicity. The unicity result is valid on a closed subset of the Diophantine torus union of full measure.

We discuss the asymptotic distribution of the directions in homology of periodic orbits of Anosov flows. We obtain a limiting measure which is either a Dirac measure on a single point or is fully supported. In the latter case, we relate the result to a more general equidistribution problem.

Given a semigroup of surjective local homeomorphisms on a compact space X we consider the corresponding semigroup of *-endomorphisms on C(X) and discuss the possibility of extending it to an interaction group, a concept recently introduced by the first named author. We also define a transformation groupoid whose C*-algebra turns out to be isomorphic to the crossed product algebra for the interaction group. Several examples are considered, including one which gives rise to a slightly different construction and should be interpreted as being the C*-algebra of a certain polymorphism.

We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangian tori by gluing together local KAM conjugacies with the help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly integrable system and an integrable one. This leads to the preservation of geometry, which allows us to define all non-trivial geometric invariants of an integrable Hamiltonian system (like monodromy) for a nearly integrable one.

We study the ergodic properties of the additive Euclidean algorithm $f$ defined in $\mathbb{R}^2_+$. A natural extension of $f$ is obtained using the action of ${\it SL}(2, \mathbb{Z})$ on a subset of ${\it SL}(2, \mathbb{R})$. We prove that, while $f$ is an ergodic transformation with an infinite invariant measure equivalent to the Lebesgue measure, the invariant measure is not unique up to scalar multiples, and in fact there is a continuous family of such measures.

We show that, for C1-generic diffeomorphisms, every chain recurrent class C that has a partially hyperbolic splitting with dimEc=1 either is an isolated hyperbolic periodic orbit, or is accumulated by non-trivial homoclinic classes. We also prove that, for C1-generic diffeomorphisms, any chain recurrent class that has a dominated splitting with dim(E)=1 either is a homoclinic class, or the bundle E is uniformly contracting. As a corollary we prove in dimension three a conjecture of Palis, which announces that any C1-generic diffeomorphism is either Morse–Smale, or has a non-trivial homoclinic class.

Let $T(1),\dots,T(d)$ be conservative, invertible, non-singular, commuting transformations on the Polish measure space $(X,m)$. Then for $f$ and $p$ in $L^1(m)$ with $p>0$,

\[ \frac{{\hat T}(1)_{-N}^N \dotsb {\hat T}(d)_{-N}^Nf}{{\hat T}(1)_{-N}^N \dotsb {\hat T}(d)_{-N}^Np}\to E[f | {\mathcal I}]/E[p| {\mathcal I}]\quad \text{as }N\to\infty. \]

We develop further our study of quasiconformal Anosov flows in our previous (Y. Fang. Smooth rigidity of uniformly quasiconformal Anosov flows. Ergod. Th. & Dynam. Sys.24 (2004), 1–23). For example, we prove the following result: Let φ be a transversely symplectic Anosov flow with dim  Ess≥2 and dim  Esu≥2. If φ is quasiconformal, then it is, up to finite covers, orbit equivalent either to the suspension of a symplectic hyperbolic automorphism of a torus or to the geodesic flow of a closed hyperbolic manifold.

Let $T$ be a finite subset of the complex unit circle ${\mathbb{S}^1}$ and define $f: {\mathbb{S}^1} \mapsto {\mathbb{S}^1}$ by $f(z) = z^d$. Let $\mathrm{CH}(T)$ denote the convex hull of $T$. If $\mathrm{card}(T) = N \geq 3$, then $\mathrm{CH}(T)$ defines a polygon with $N$ sides. The $N$-gon $\mathrm{CH}(T)$ is called a wandering N-gon if for every two non-negative integers $i \neq j,\ \mathrm{CH}(f^i(T))$ and $\mathrm{CH}(f^j(T))$ are disjoint $N$-gons. A non-degenerate chord of ${\mathbb{S}^1}$ is said to be critical if its two endpoints have the same image under $f$. Then for a critical chord, it is natural to define its (forward) orbit by the forward iterates of the endpoints. Similarly, we call a critical chord recurrent if one of its endpoints is recurrent under $f$. The main result of our study is that a wandering $N$-gon has at least $N-1$ recurrent critical chords in its limit set (defined in a natural way) having pairwise disjoint, infinite orbits. Using this result, we are able to strengthen results of Blokh, Kiwi, Levin and Thurston about wandering polygons of laminations. We also discuss some applications to the dynamics of polynomials. In particular, our study implies that if $v$ is a wandering non-precritical vertex of a locally connected polynomial Julia set, then there exists at least $\mathrm{ord}(v) -1$ recurrent critical points with pairwise disjoint orbits, all having the same $\omega$-limit set as $v$. Thus, we likewise strengthen results about wandering vertices of polynomial Julia sets.

In this paper some families of skew product self-maps $F$ on the square are considered. The main example is a family forming a two-dimensional analogue of the tent map family. According to the assumptions made in this paper these maps are almost injective. This means that the points of the attractor having more than one inverse image form a set of measure zero for all interesting measures. It may be that $F$ does not have a finite Markov partition. The Hausdorff dimension of the attractor is computed.

Averages of functionals along trajectories are studied by evaluating the averages along chains. This yields results for the possible limits and, in particular, for ergodic limits. Applications to Lyapunov exponents and to concepts of rotation numbers of linear Hamiltonian flows and of general linear flows are given.

We show that any probability preserving transformation of a metric space is mixing as soon as there are no non-constant $L^2$-functions which are invariant under both the stable and unstable distributions. This generalizes a result of Babillot.

For polynomials $f$ on the complex plane with a dendrite Julia set we study invariant probability measures, obtained from a reference measure. To do this we follow Keller [K1] in constructing canonical Markov extensions. We discuss ‘liftability’ of measures (both $f$-invariant and non-invariant) to the Markov extension, showing that invariant measures are liftable if and only if they have a positive Lyapunov exponent. We also show that $\delta$-conformal measure is liftable if and only if the set of points with positive Lyapunov exponent has positive measure.

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.

We augment the conjugation approximation method with explicit estimates on the conjugacy map. This allows us to construct ergodic volume-preserving diffeomorphisms measure-theoretically isomorphic to any a priori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.

A reverse weak type inequality for the conjugate function is generalized to the ergodic case. The inequality allows us to prove the Riesz–Zygmund theorem (if $f\geq 0$ and $f,\widetilde{f}\in L$, then $f\in L\log L$) for the ergodic Hilbert transform.

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.

Our goal in this paper is to give an example of a one-parameter family of rational maps for which, in the parameter plane, there is a Cantor set of simple closed curves consisting of parameters for which the corresponding Julia set is a Sierpiński curve. Hence, the Julia sets for each of these parameters are homeomorphic. However, each of the maps in this set is dynamically distinct from (i.e. not topologically conjugate to) any other map in this set (with only finitely many exceptions). We also show that, in the dynamical plane for any map drawn from a large open set in the connectedness locus in this family, there is a Cantor set of invariant simple closed curves on which the map is conjugate to the product of certain subshifts of finite type with the maps on the unit circle.

We study curves of fixed points for certain diffeomorphisms of ${\mathbb{R}}^3$ that are induced by automorphisms of a trace algebra. We classify these curves. There is a function $E$ which is invariant under all such trace maps and the level surfaces $E_t: E=t$ are invariant; a point of $E_t$ will be said to have level $t$. The surface $E_1$ is significant. Then most fixed points on $E_1$ are actually on a curve $\gamma$ of fixed points interior to $E_1$. We describe the possibilities for the other end of $\gamma$ on $E_1$.