We show that, providing a metric space $X$ has a boundary that is in somesense similar to the boundary of hyperbolic space, the iterates of acontraction $f:X\to X$ converge locally uniformly to a point in, or on theboundary of, $X$. This generalises the Denjoy–Wolff theorem for analyticself-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 contractionof $D$ 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 restrictedto the omega limit set $X$ of the critical point for cases where $X$ is aCantor set.In particular, many cases where $X$ isa measure attractor of $f$ are included. We give two classes of examples ofsuch maps, both generalizing unimodal Fibonacci maps [LM,BKNS]. In allcases $f_{|X}$ is a continuous factor of a generalized odometer (an addingmachine-like dynamical system), and at the same time $f_{|X}$ factors ontoan irrational circle rotation. In some of the examples we obtain irrationalrotations 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 ofiterates $(f^n)$ is not normal in any neighbourhood of $z_0$. In thispaper, 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 ofpolynomialsof degree at least two. If we can find a hyperbolic domain $M$ whichis invariant under all $f_n$, $n\in\N$, $\infty\in M$ and$F_n\to\infty\ (n\to\infty)$ locally uniformly in $M$, then weask whether these sets of non-normality,which we will also call Julia sets, have properties which we know from theclassical 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 (whereno domain $M$ exists) which have a pathological behaviour in contrast to theclassical case.

We find conditions on the ratios of dissection of a Cantor set so that thegroup it generates under addition has positive Lebesgue measure. Inparticular, we answer affirmatively a special case of a conjecture posed byJ. Palis.

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

For a class of multiphase averaging problems inwhich the unperturbed flow of the fast variables is a transitive Anosov flowor is ‘sufficiently rapidly mixing’, we obtain what we believe isan optimalpower-law estimate, in the small parameter, of the rate at which the averagemaximum difference between solutions of the exact and averaged problemsconverges to zero. This verifies and extends an earlier conjectural remark byV. I. Arnold.

We show that the geodesic flow of a metricall of whose geodesics are closed is completely integrable,with tame integrals of motion. Applicationsto 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 isometriesis completelyintegrable by tame integrals.

Let $G={\rm SL_2({\bf R})}$ (or $G={\rm SO}(n,1)$) act ergodically on aprobability 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 surfaceis 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 examplesin which it is not Bernoullian.

We consider $g$-measures for the one-sided shifton a sequence space.We give a condition which guarantees thatthe $g$-measure is unique andhas a Bernoulli natural extension.

In this paper we consider the class ofsurface mappings consisting of those maps which have the least number offixed points possible among all maps in their homotopy class.When the surface has non-empty boundary, we show that for mappingsin 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 apermutation called its type. If this permutation is a cycle, it is calledits orbit type. It has been shown by Geller and Tolosa thatMisiurewicz–Nitecki orbit types of period $n$ congruent to $1$ (mod 4)andtheirgeneralizations to orbit types of period $n$ congruent to $3$(mod 4) havemaximal entropy among all orbit types of odd period $n$, and indeedamongall permutations of period $n$. We further generalize this family topermutations of even period $n$ and show that they again attain maximalentropyamongst $n$-permutations.

Let $T:X\to{\Bbb R}$ be a piecewisemonotonic map, where $X$ is a finite union of closedintervals. Define $R(T)=\bigcap_{n=0}^{\infty}\overline{T^{-n}X}$, and suppose that $(R(T),T)$ hasa unique maximal measure $\mu$. The influence ofsmall perturbations of $T$ on the maximal measure isinvestigated. If $(R(T),T)$ has positive topologicalentropy, and if a certain stability condition issatisfied, then every piecewise monotonic map$\tilde{T}$, which is contained in a sufficientlysmall neighbourhood of $T$, has a unique maximalmeasure $\tilde{\mu}$, and the map$\tilde{T}\mapsto\tilde{\mu}$ is continuousat $T$.

We consider one-dimensional flows which arise as hyperbolicinvariant sets of a smooth flow on a manifold. Included in our data is thetwisting in the local stable and unstable manifolds. A topologicalinvariant 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 recurrentcritical 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 $m$. Themeasure $\mu$ is then proved to be ergodic and conservative and we study theset of those points whose all openneighborhoods haveinfinite measure $\mu$. Developing the concept of the inverse jumptransformation we show that the packing and Hausdorffdimensions of the conformal measure are equal to $h$. We also provide somesufficient conditions for Hausdorff and box dimensionsof 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 exchangetransformation is uniquely ergodic. Hence, the ergodic theorem predicts thatthe number $\chi_i(x,n)$ of visits of our trajectory to the $i$th subintervalwould be approximately $\lambda_i n$. Here $\lambda_i$ is the length of thecorresponding subinterval of our unit interval $X$. In this paper we give anestimate for the deviation of the actual number of visits to the $i$thsubinterval $X_i$ from one predicted by the ergodic theorem.

We prove that for almost all interval exchangetransformations 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 $n^{\theta_2/\theta_1}$. Thenumbers $0\le \theta_2 < \theta_1$ depend only on the permutation $\pi$corresponding to the interval exchange transformation (actually, only on theRauzy class of the permutation). In the case of interval exchange of twointervals we obviously have $\theta_2=0$. In the case of exchange of threeand more intervals the numbers $\theta_1, \theta_2$ are the two top Lyapunovexponents related to the corresponding generalized Gauss map on the space ofinterval exchange transformations.

The limit above ‘converges to the bound’ uniformly forall $x\in X$ in the following sense. For any $\varepsilon >0$ the ratioof 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 thestarting point $x\in X$.