We show that, providing a metric space $X$ has a boundary that is in some sense similar to the boundary of hyperbolic space, the iterates of a contraction $f:X\to X$ converge locally uniformly to a point in, or on the boundary of, $X$. This generalises the Denjoy–Wolff theorem for analytic self-maps of the unit disc in the complex plane, and also shows that if $D$ is a bounded strictly convex subdomain of ${\Bbb R}^n$, then any contraction of $D$ with respect to the Hilbert metric of $D$ converges to a point in the closure of $D$.

We investigate the dynamics of unimodal maps $f$ of the interval restricted to the omega limit set $X$ of the critical point for cases where $X$ is a Cantor set. In particular, many cases where $X$ is a measure attractor of $f$ are included. We give two classes of examples of such maps, both generalizing unimodal Fibonacci maps [LM, BKNS]. In all cases $f_{|X}$ is a continuous factor of a generalized odometer (an adding machine-like dynamical system), and at the same time $f_{|X}$ factors onto an irrational circle rotation. In some of the examples we obtain irrational rotations on more complicated groups as factors.

In the classical iteration theory we say that for a given polynomial $f$ a point $z_0\in\C$ belongs to the Julia set if the sequence of iterates $(f^n)$ is not normal in any neighbourhood of $z_0$. In this paper, we look at the set of non-normality of $(F_n)$, $F_n:=f_n\circ\cdots\circ f_1$, where $(f_n)$ is a given sequence of polynomials of degree at least two. If we can find a hyperbolic domain $M$ which is invariant under all $f_n$, $n\in\N$, $\infty\in M$ and $F_n\to\infty\ (n\to\infty)$ locally uniformly in $M$, then we ask whether these sets of non-normality, which we will also call Julia sets, have properties which we know from the classical case. We show that the Julia set is self-similar. Furthermore, the Julia set is perfect or finite. The finite case may actually occur. We will also give some sufficient conditions for the Julia set being perfect. In the last section we give some examples of sequences of polynomials (where no domain $M$ exists) which have a pathological behaviour in contrast to the classical case.

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.

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.

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 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.

Let $G={\rm SL_2({\bf R})}$ (or $G={\rm SO}(n,1)$) act ergodically on a probability space $(X,m)$. We consider the ergodic properties of the flow $(X,m,\{g_t\})$, where $\{g_t\}$ is a Cartan subgroup of $G$. The geodesic flow on a compact Riemann surface is an example of such a flow; here $X=G/\Gamma$ is a transitive $G$-space, $G={\rm SL_2({\bf R})}$ and $\Gamma\subset G$ is a lattice. In this case the flow is Bernoullian.

For the general ergodic $G$-action, the flow $(X,m,\{g_t\})$ is always a $K$-flow, however there are examples in which it is not Bernoullian.

We consider $g$-measures for the one-sided shift on a sequence space. We give a condition which guarantees that the $g$-measure is unique and has a Bernoulli natural extension.

In this paper we consider the class of surface mappings consisting of those maps which have the least number of fixed points possible among all maps in their homotopy class. When the surface has non-empty boundary, we show that for mappings in this class the index of a fixed point is bounded above by $1$ and below by $2\chi -1$. This generalizes a well known result for pseudo-Anosov homeomorphisms. A proof of a Jiang–Guo type inequality is also given.

A finite invariant set of a continuous map of an interval induces a permutation called its type. If this permutation is a cycle, it is called its orbit type. It has been shown by Geller and Tolosa that Misiurewicz–Nitecki orbit types of period $n$ congruent to $1$ (mod 4) and their generalizations to orbit types of period $n$ congruent to $3$ (mod 4) have maximal entropy among all orbit types of odd period $n$, and indeed among all permutations of period $n$. We further generalize this family to permutations of even period $n$ and show that they again attain maximal entropy amongst $n$-permutations.

Let $T:X\to{\Bbb R}$ be a piecewise monotonic map, where $X$ is a finite union of closed intervals. Define $R(T)=\bigcap_{n=0}^{\infty} \overline{T^{-n}X}$, and suppose that $(R(T),T)$ has a unique maximal measure $\mu$. The influence of small perturbations of $T$ on the maximal measure is investigated. If $(R(T),T)$ has positive topological entropy, and if a certain stability condition is satisfied, then every piecewise monotonic map $\tilde{T}$, which is contained in a sufficiently small neighbourhood of $T$, has a unique maximal measure $\tilde{\mu}$, and the map $\tilde{T}\mapsto\tilde{\mu}$ is continuous at $T$.

We consider one-dimensional flows which arise as hyperbolic invariant sets of a smooth flow on a manifold. Included in our data is the twisting in the local stable and unstable manifolds. A topological invariant sensitive to this twisting is obtained.

Let $h$ be the Hausdorff dimension of the Julia set of a rational function $T$ with no non-periodic recurrent critical points and let $m$ be the only $h$-conformal measure for $T$. We prove the existence of a $\sigma$-finite $T$-invariant measure $\mu$ equivalent with $m$. The measure $\mu$ is then proved to be ergodic and conservative and we study the set of those points whose all open neighborhoods have infinite measure $\mu$. Developing the concept of the inverse jump transformation we show that the packing and Hausdorff dimensions of the conformal measure are equal to $h$. We also provide some sufficient conditions for Hausdorff and box dimensions of the Julia set to be equal.

Consider a long piece of a trajectory $x, T(x), T(T(x)), \ldots, T^{n-1}(x)$ of an interval exchange transformation $T$. A generic interval exchange transformation is uniquely ergodic. Hence, the ergodic theorem predicts that the number $\chi_i(x,n)$ of visits of our trajectory to the $i$th subinterval would be approximately $\lambda_i n$. Here $\lambda_i$ is the length of the corresponding subinterval of our unit interval $X$. In this paper we give an estimate for the deviation of the actual number of visits to the $i$th subinterval $X_i$ from one predicted by the ergodic theorem.

We prove that for almost all interval exchange transformations the following bound is valid: $$ \max_{\ssty x\in X \atop \ssty 1\le i\le m} \limsup_{n\to +\infty} \frac {\log | \chi_i(x,n) -\lambda_in|}{\log n} = \frac{\theta_2}{\theta_1} < 1. $$ Roughly speaking the error term is bounded by $n^{\theta_2/\theta_1}$. The numbers $0\le \theta_2 < \theta_1$ depend only on the permutation $\pi$ corresponding to the interval exchange transformation (actually, only on the Rauzy class of the permutation). In the case of interval exchange of two intervals we obviously have $\theta_2=0$. In the case of exchange of three and more intervals the numbers $\theta_1, \theta_2$ are the two top Lyapunov exponents related to the corresponding generalized Gauss map on the space of interval exchange transformations.

The limit above ‘converges to the bound’ uniformly for all $x\in X$ in the following sense. For any $\varepsilon >0$ the ratio of logarithms would be less than $\theta_2(\pi)/\theta_1(\pi)+\varepsilon $ for all $n\ge N(\varepsilon)$, where $N(\varepsilon)$ does not depend on the starting point $x\in X$.