Let $f : V \to U$ be a (generalized) polynomial-like map. Suppose that harmonic measure $\omega= \omega(\cdot,\infty)$ on the Julia set $J_{f}$ is equal to the measure of maximal entropy $m$ for $f : J_{f} \hookleftarrow$. Then the dynamics $(f,V,U)$ is called maximal. We are going to give a criterion for the dynamics to be conformally equivalent to a maximal one, that is to be conformally maximal. In the second part of this paper we construct an invariant ‘harmonic’ measure $\mu$ such that ${d\mu}/{d\omega}$ is Hölder for certain dynamics. This allows us to prove in this class of dynamical systems that $\omega\approx m$ is necessary and sufficient for $(f,V,U)$ to be conformally maximal. In the particular case when $f$ is expanding and $J_{f}$ is a circle, our result becomes a theorem of Shub and Sullivan; so throughout the paper we are dealing with an analog of a theorem of Shub and Sullivan on ‘wild’ (e.g. totally disconnected) $J_{f}$ and for certain non-expanding $f$. We also construct (under certain assumptions) invariant harmonic measure on $J_{f}$. In this respect, our work stems from one of the works of Carleson.

The variational principle states that the topological entropy of a topological dynamical system is equal to the sup of the entropies of invariant measures. It is proved that for any finite open cover there is an invariant measure such that the topological entropy of this cover is less than or equal to the entropies of all finer partitions. One consequence of this result is that for any dynamical system with positive topological entropy there exists an invariant measure whose set of entropy pairs is equal to the set of topological entropy pairs.

We introduce entropy convergence rates as isomorphism invariants for measure-preserving systems and prove several general facts concerning these rates for aperiodic systems, completely ergodic systems and rank-one systems. We will for example show that for any completely ergodic system $(X,T)$ and any non-trivial partition $\alpha$ of $X$ into two sets we have $\limsup_{n\rightarrow\infty}H(\alpha_0^{n-1})/g(\log_2n)=\infty$, whenever $g$ is a positive increasing function on $(0,\infty)$ such that $g(x)/x^2$ is integrable.

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.

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 $1/t$.

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.

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 $f_{\beta,\alpha}$.

It is an open problem to determine for which maps $f$, any compact invariant set $K$ carries an ergodic invariant measure of the same Hausdorff dimension as $K$. If $f$ is conformal and expanding, then it is a known consequence of the thermodynamic formalism that such measures do exist. (We give a proof here under minimal smoothness assumptions.) If $f$ has the form $f(x_1,x_2)=(f_1(x_1),f_2(x_2))$, where $f_1$ and $f_2$ are conformal and expanding maps satisfying $\inf \vert Df_1\vert\geq\sup\vert Df_2\vert$, then for a large class of invariant sets $K$, we show that ergodic invariant measures of dimension arbitrarily close to the dimension of $K$ do exist. The proof is based on approximating $K$ by self-affine sets.

We present an optimal result for the regularity of the invariant distributions of an Anosov system in terms of expansion and contraction rates. Compared to earlier results it avoids an ‘infinitesimal loss’ in the Hölder exponent.

The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by Denjoy [1]. In [5] one of us gave a new criterion. There is an example satisfying Denjoy's bounded variation condition rather than the Zygmund condition of [5], and vice versa. This paper will give the third criterion which is implied by either of the above criteria.

In this paper we show that the geodesic flow in a Hedlund-type metric on the 3-torus possesses the shadowing property. This implies, in particular, that any rotation vector is represented by a geodesic, a fact that in the two-dimensional case is given by the Aubry–Mather theory, while in the higher-dimensional case is still unknown.

We consider the subset of the Julia set called the residual Julia set, which comes from an analogy of the residual limit set of a Kleinian group. We give a necessary and sufficient condition in order that the residual Julia set is empty in the case of hyperbolic rational functions.

Let $M$ be a compact Riemannian manifold with no conjugate points such that its geodesic flow is expansive. We show that there exists a local product structure in the unit tangent bundle of the manifold which is invariant under the geodesic flow. In particular, we have that the set of closed geodesics is dense and that the flow is topologically transitive.

In this paper we study the set of bounded geodesics on a general, geometrically finite $(N+1)$-manifold of constant negative curvature. We obtain the result that the Hausdorff dimension of this set is equal to $2 \delta$, where $\delta$ denotes the exponent of convergence of the associated Kleinian group. The proof of this shows, in particular, that if the group has parabolic elements, then the set of limit points which are badly approximable with respect to the parabolic fixed points has Hausdorff dimension equal to $\delta$.

Green [5] conjectured that if $M$ is a closed Riemannian manifold of negative sectional curvature such that the mean curvatures of the horospheres through each point depend only on the point, then $V$ is a locally symmetric space of rank one. He proved this in dimension two. In this paper we prove that under Green's assumption, $M$ must be asymptotically harmonic and that the geodesic flow on $M$ is $C^{\infty}$ conjugate to that of a locally symmetric space of rank one. Combining this with the recent rigidity theorem of Besson–Courtois–Gallot [1], it follows that Green's conjecture is true for all dimensions.