Consider a one-parameter family of circle diffeomorphisms which unfolds a saddle-node periodic orbit at the edge of an ‘Arnold tongue’. Recently it has been shown that homoclinic orbits of the saddle-node periodic points induce a ‘transition map’ which completely describes the smooth conjugacy classes of such maps and determines the universalities of the bifurcations resulting from the disappearance of the saddle-node periodic points. We show that after the bifurcation the relative density (measure) of parameter values corresponding to irrational rotation numbers is completely determined by the transition map and give a formula for this density. It turns out that this density is always less than 1 and generically greater than 0, with the exceptional cases having infinite co-dimension.

We give an asymptotic estimate for the number of periodic orbits of an Anosov flow which are subject to multi-dimensional constraints. We also study their spatial distribution. For instance, we describe the distribution of periodic orbits with respect to homology classes, for both homologically full Anosov flows and suspensions of Anosov transformations.

We prove that if $F$ is an orientation-preserving homeomorphism of the plane that leaves invariant an irreducible plane separating continuum $\Delta$, then, with the possible exception of three numbers, if $p/q$ is a reduced rational in the interior of the convex hull of the rotation set of $F\vert_{\Delta}$ (with respect to some lift) there are at least two distinct periodic orbits of $F\vert_{\Delta}$ of period $q$ and rotation number $p/q$. This result also applies to certain nonseparating invariant continua.

This paper investigates the dynamical properties of a class of urn processes and recursive stochastic algorithms with constant gain which arise frequently in control, pattern recognition, learning theory, and elsewhere.

It is shown that, under suitable conditions, invariant measures of the process tend to concentrate on the Birkhoff center of irreducible (i.e. chain transitive) attractors of some vector field $F: {\Bbb R}^d \rightarrow {\Bbb R}^d$ obtained by averaging. Applications are given to simple situations including the cases where $F$ is Axiom A or Morse–Smale, $F$ is gradient-like, $F$ is a planar vector field, $F$ has finitely many alpha and omega limit sets.

For every Pisot number $\theta$, and every real number $\beta$ greater than one not equivalent to $\theta$, if $\mu$ is a Bernoulli measure relative to the $\theta$-transformation, then for $\mu$-almost all $x$ the sequence $(x\beta^n)_{n\geq 0}$ is uniformly distributed modulo one.

Techniques from the authors' previous paper [BD1] are applied to unique $G$-measures, where the $g$-functions are uniformly Lipschitz. Conditions are found when such a measure is, and is not, equivalent to a measure of product type.

We consider the entropy of systems of random transformations, where the transformations are chosen from a set of generators of a ${\Bbb Z}^d $ action. We show that the classical definition gives unsatisfactory entropy results in the higher-dimensional case, i.e. when $d \geq 2$. We propose a definition of the entropy for random group actions which agrees with the classical definition in the one-dimensional case, and which gives satisfactory results in higher dimensions. This definition is based on the fibre entropy of a certain skew product. We identify the entropy by an explicit formula which makes it possible to compute the entropy in certain cases.

Let ${\cal R}$ be an ergodic discrete equivalence relation on a Lebesgue space and $\alpha$ its cocycle with values in a locally compact group $G$. We say that an automorphism of ${\cal R}$ is compatible with $\alpha$ if it preserves the cohomology class of $\alpha$. We introduce a quasi-order relation on the set of all cocycles of ${\cal R}$ by means of comparison of the corresponding groups of all automorphisms being compatible with them. We find simple necessary and sufficient conditions under which two cocycles of a hyperfinite measure-preserving equivalence relation with values in compact (possibly different) groups are connected by this relation. Next, given an ergodic subrelation ${\cal S}$ of ${\cal R}$, we investigate the problem of extending ${\cal S}$-cocycles up to ${\cal R}$-cocycles and improve the recent results of Gabriel–Lemańczyk–Schmidt. As an application we study the problem of lifting of automorphisms of ${\cal R}$ up to automorphisms of the skew product ${\cal R}\times_\alpha G$, provide new short proofs of some known results and answer several questions from [ALV, ALMN].

Let $F_\lambda(z)= \lambda z + {\cal O}(z^2)$ be a one parameter holomorphic family of holomorphic maps defined in a neighborhood of $\lambda_0* \overline{\mbox{D}}$. Assume that $F_{\lambda_0}$ has a Siegel disc $\Delta_0$, then the orbit of a point in the interior of the Siegel disc can be followed very closely by periodic orbits of nearby maps. The same technique is applied to Herman rings and Cremer points.

We define 2-critically finite maps of ${\bf P}^2$ and show that they have no nontrivial closed backward invariant sets. In particular, their Julia sets $J_1$, defined as the support of a natural invariant measure, are equal to ${\bf P}^2$. We also prove that repelling periodic points are dense for such maps.

We have two main purposes in this paper. One is to give some sufficient conditions for the Julia set of a transcendental entire function $f$ to be connected or to be disconnected as a subset of the complex plane ${\Bbb C}$. The other is to investigate the boundary of an unbounded periodic Fatou component $U$, which is known to be simply-connected. These are related as follows: let $\varphi : {\Bbb D} \longrightarrow U$ be a Riemann map of $U$ from a unit disk ${\Bbb D}$, then under some mild conditions we show that the set $\Theta_{\infty}$ of all angles where $\varphi$ admits the radial limit $\infty$ is dense in $\partial {\Bbb D}$ if $U$ is an attracting basin, a parabolic basin or a Siegel disk. If $U$ is a Baker domain on which $f$ is not univalent, then $\Theta_{\infty}$ is dense in $\partial {\Bbb D}$ or at least its closure $\overline{\Theta_{\infty}}$ contains a certain perfect set, which means the boundary $\partial U$ has a very complicated structure. In all cases, this result leads to the disconnectivity of the Julia set $J_f$ in ${\Bbb C}$. If $U$ is a Baker domain on which $f$ is univalent, however, we shall show by giving an example that $\partial U$ can be a Jordan arc in ${\Bbb C}$, which has a rather simple structure, and, moreover, $J_f$ can be connected.

We also consider the connectivity of the set $J_f \cup \{ \infty \}$ in the Riemann sphere $\widehat{{\Bbb C}}$ and show that $J_f \cup \{ \infty \}$ is connected if and only if $f$ has no multiply-connected wandering domains.

In 1971, M. M. Peixoto [15] introduced an important topological invariant of Morse–Smale flows on surfaces, which he called a distinguished graph $X^*$ associated with a given flow. Here we show how the Peixoto invariant can be essentially simplified and revised by adopting a purely topological point of view connected with the embeddings of arbitrary graphs into compact surfaces. The newly obtained invariant, $X^R$, is a rotation of a graph $X$ generated by a Morse–Smale flow. (A rotation $R$ is a cyclic order of edges given in every vertex of $X$.) The invariant $X^R$ ‘reads-off’ the topological information carried by a flow, being in a one-to-one correspondence with the topological equivalence classes of Morse–Smale flowsAnd foliations, see [3]. We do not treat the case of foliations, bearing in mind that they are defined by involutive flows on covering manifolds [9].. As a counterpart to the equivalence result we prove a realization theorem for an ‘abstractly given’ $X^R$. (Our methods are completely different from those of Peixoto and they clarify the connections between graphs and flows on surfaces.) The idea of ‘rotation systems’ on graphs can be further exploited in the study of recurrent flows (and foliations) with several disjoint quasiminimal sets on surfaces [10].

We apply Thurston's characterization of postcritically finite rational maps as branched coverings of the sphere to give new classes of combination theorems for postcritically finite rational maps. Our constructions increase the degree of the map but always yield branched coverings which are equivalent to rational maps, independent of the combinatorics of the original map. The main tool is a general theorem based on the intersection number of arcs and curves which controls the region in the sphere in which an obstruction may reside.

A class of negative definite kernels is defined in terms of measure spaces. Using this concept, property (T) for a countable group $\Gamma$ is characterized in terms of measure-preserving actions of $\Gamma$, as follows. If a set $S$ is translated a finite amount by any fixed element of $\Gamma$, then there is a uniform bound on how far $S$ is translated.