Let $F$ be an area-preserving homeomorphism of the two-dimensional torus homotopic to the identity and let $f$ be a lift to the plane which preserves the center of mass. We prove that $f$ has at least three fixed points whose images are distinct in the torus.

Let $f$ be a flip-invariant Hölder continuous function on the unit tangent bundle $T^1 M$ of a closed negatively curved Riemannian manifold $M$. We show that conditionals on strong unstable manifolds of the Gibbs equilibrium state defined by $f$ can be realized as Hausdorff measures. Moreover, cohomology classes of flip invariant cocycles are in one-to-one correspondence to cross ratios on the space of four pairwise distinct points of the ideal boundary of the universal covering $\tilde M$ of $M$.

The closing lemma is proved for various classes of holomorphic self-maps on ${\Bbb C}^k$. These classes include those consisting of biholomorphic maps, endomorphisms and symplectomorphisms.

We find conditions on the ratios of dissection of a Cantor set so that the group it generates under addition has positive Lebesgue measure. In particular, we answer affirmatively a special case of a conjecture posed by J. Palis.

The two-dimensional analogue of the Sharkovski order on periods for maps of the interval restricts to a partial order on essential pseudo-Anosov conjugacy classes in the mapping class group of the $n$-times punctured disk. In this paper we give an explicit description of this restricted partial order in the case when $n=3$.

Let $S(N)$ be a random walk on a countable abelian group $G$ which acts on a probability space $E$ by measure-preserving transformations $(T_v)_{v\in G}$. For any $\Lambda \subset E$ we consider the random return time $\tau$ at which $T_{S(\tau)}\in\Lambda$. We show that the corresponding induced skew product transformation is K-mixing whenever a natural subgroup of $G$ acts ergodically on $E$.

There exist uniquely ergodic bijective affine interval exchange transformations of $[0,1)$ having wandering intervals and such that the support of the invariant measure is a Cantor set.

We find analytic solutions $\varphi_{t}$ to the conjugation equation $$ f_{t} \circ \varphi_{t}=\varphi_{t} \circ f_{0}, $$ where $f_{t}$ and $\varphi_{t}$ are analytic diffeomorphisms from ${\Bbb T}^{n}$ to itself that depend analytically on $t$, and $f_{0}$ is an Anosov toral diffeomorphism with reduced spectrum and which is diagonalizable.

We consider the ergodic skew product of translations on the torus ${\Bbb T}^{2}$. We characterize a class of smooth functions with exponential correlation decay rate and a class of subsets with an exponential rate of mixing. We show that the class of functions with sub-exponential correlation decay rate is generic. We also characterize another class of subsets with mixing rate of the order of $1/t$.

In [R1] a notion of restricted orbit equivalence for ergodic transformations was developed. Here we modify that structure in order to generalize it to actions of higher-dimensional groups, in particular ${\Bbb Z}^d$-actions. The concept of a ‘size’ is developed first from an axiomatized notion of the size of a permutation of a finite block in ${\Bbb Z}^d$. This is extended to orbit equivalences which are cohomologous to the identity and, via the natural completion, to a notion of restricted orbit equivalence. This is shown to be an equivalence relation. Associated to each size is an entropy which is an equivalence invariant. As in the one-dimensional case this entropy is either the classical entropy or is zero. Several examples are discussed.

For a class of multiphase averaging problems in which the unperturbed flow of the fast variables is a transitive Anosov flow or is ‘sufficiently rapidly mixing’, we obtain what we believe is an optimal power-law estimate, in the small parameter, of the rate at which the average maximum difference between solutions of the exact and averaged problems converges to zero. This verifies and extends an earlier conjectural remark by V. I. Arnold.

We study the topological entropy of a class of transformations with mild singularities: the generalized polygon exchanges. This class contains, in particular, polygonal billiards. Our main result is a geometric estimate, from above, on the topological entropy of generalized polygon exchanges. One of the applications of our estimate is that the topological entropy of polygonal billiards is zero. This implies the subexponential growth of various geometric quantities associated with a polygon. Other applications are to the piecewise isometries in two dimensions, and to billiards in rational polyhedra.

We construct a universal Cuntz–Krieger algebra ${\cal {AO}}_A$, which is isomorphic to the usual Cuntz–Krieger algebra ${\cal O}_A$ when $A$ satisfies condition $(I)$ of Cuntz and Krieger. The Cuntz classification of ideals in ${\cal O}_A$ when $A$ satisfies condition $(II)$ extends to a classification of the gauge invariant ideals in ${\cal {AO}}_A$. We use this to describe the topology on the primitive ideal space of ${\cal {AO}}_A$.

In this paper we study a new functional on the space of metrics with negative curvature on a compact manifold. It is a linear combination of Liouville entropy and total scalar curvature. Locally symmetric spaces are critical points of this functional. We provide an explicit formula for its second derivative at metrics of constant negative curvature. In particular, this shows that a metric of constant curvature is a local maximum.

We study a qualitative theory of compact, transversely piecewise-smooth foliated $S^1$-bundles. We show that it has the same qualitative properties as that of smooth codimension-one foliations on a compact manifold. By the obtained qualitative properties, we can deduce that the expansion growth of this foliation is classified and that the entropy of this foliation is positive if and only if there exists a resilient leaf.

We consider Anosov flows in 3-manifolds. Suppose that there is a rank-two free abelian subgroup of the fundamental group of the manifold, so that none of its elements can be represented by a closed orbit of the flow. We then show that the flow is topologically conjugate to a suspension of an Anosov diffeomorphism. As a consequence we prove that if $T$ is an incompressible torus so that no loop in $T$ is freely homotopic to a closed orbit of the flow, then $T$ is isotopic to a transverse torus. Finally, we show that if $T$ is an incompressible torus transverse to the stable foliation, then either there is a closed leaf in the induced foliation in $T$, or the flow is topologically conjugate to a suspension Anosov flow.

Let $T$ be a locally finite simplicial tree and let $\Gamma\subset{\rm Aut}(T)$ be a finitely generated discrete subgroup. We obtain an explicit formula for the critical exponent of the Poincaré series associated with $\Gamma$, which is also the Hausdorff dimension of the limit set of $\Gamma$; this uses a description due to Lubotzky of an appropriate fundamental domain for finite index torsion-free subgroups of $\Gamma$. Coornaert, generalizing work of Sullivan, showed that the limit set is of finite positive measure in its dimension; we give a new proof of this result. Finally, we show that the critical exponent is locally constant on the space of deformations of $\Gamma$.

We introduce transition entropy and periodic entropy for locally compact subshifts. Finiteness of both characterizes the existence of a finite generator. Finiteness of the transition entropy characterizes the existence of a generator with bounded degree. We extend Krieger's embedding theorem and characterize the locally compact non-dense subsystems of a (compact) mixing shift of finite type.

Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic on the unit disk $|z|<1$ and analytic on $|z|<1/\beta$. This paper shows that the singularities of $L_{\beta,\alpha}(z)$ on the circle $|z|=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N):0\le l\le N-1\}$, for some integer $N\ge 1$. Here $N$ can be taken to be the period $N_{\beta,\alpha}$ of a certain Markov chain $\Sigma_{\beta,\alpha}$ which encodes information about generalized lap numbers $L_{n}(i,j)$ of $f_{\beta,\alpha}$, where $L_{n}(i,j)$ counts monotonic pieces of $f_{\beta,\alpha}^{n}$ whose image is $[f^{i}(0),f^{j}(1^{-}))$. We show that $N_{\beta,\alpha}=1$ whenever $\beta>2$. Finally, we give the criterion that $N_{\beta,\alpha}=1$ if and only if for all $n\ge 1$ the map $f_{\beta,\alpha}^{n}$ is ergodic with respect to the maximal entropy measure of $f_{\beta,\alpha}$.

We show that the geodesic flow of a metric all of whose geodesics are closed is completely integrable, with tame integrals of motion. Applications to classical examples are given; in particular, it is shown that the geodesic flow of any quotient $M/\Gamma$ of a compact, rank one symmetric space $M$ by a finite group acting freely by isometries is completely integrable by tame integrals.