We study Anosov diffeomorphisms on surfaces with small holes. The points that are mapped into the holes disappear and never return. In our previous paper we proved the existence of a conditionally invariant measure $\mu_+$. Here we show that the iterations of any initially smooth measure, after renormalization, converge to $\mu_+$. We construct the related invariant measure on the repeller and prove that it is ergodic and K-mixing. We prove the escape rate formula, relating the escape rate to the positive Lyapunov exponent and the entropy.

Given a continuous map on a locally compact metric space and an isolating neighborhood which is decomposed into two disjoint isolating neighborhoods, it is shown that the spectral information of the associated Conley indices is sufficient to conclude the existence of a semi-conjugacy onto the full shift dynamics on two symbols.

We consider deterministic perturbations $\ddot q^\varepsilon(t)+F'(q^\varepsilon(t))=\varepsilon b(\dot q^\varepsilon(t),q^\varepsilon(t))$ of an oscillator $\ddot q+F'(q)=0$, $q\in{\mathbb R}^1$. Assume that $\lim_{|q|\to\infty}F(q)=\infty$ and that $F'(q)$ has a finite number of nondegenerate zeros. For a generic $F$, if $\partial b/\partial\dot q<0$ (as in the case of friction), then typical orbits are attracted to points where $F$ has a local minimum. For $0<\varepsilon\ll1$, the equilibrium to which the trajectory is attracted is ‘random’. To study this randomness, which is caused by the sensitive behavior of trajectories near the saddle points, we consider the graph $\Gamma$ homeomorphic to the space of connected components of the level sets of the Hamiltonian $H(p,q)=p^2/2+F(q)$. We show that, as $\varepsilon\to0$, the slow component of $(p^\epsilon(t/\epsilon),q^\epsilon(t/\epsilon))$ tends to a certain stochastic process on $\Gamma$ which is deterministic inside the edges and branches at the interior vertices into adjacent edges with probabilities which can be calculated through the Hamiltonian $H$ and the perturbation $b$.

Our goal in this paper is to describe the structure of the Julia set of complex exponential functions that possess an attracting cycle. When the cycle is a fixed point, it is known that the Julia set is a ‘Cantor bouquet’, a union of uncountably many distinct curves or ‘hairs’. When the period of the cycle is greater than one, infinitely many of the hairs in the bouquet become pinched or attached together. In this paper, we develop an algorithm to determine which of these hairs are attached. Of crucial importance in this construction is the kneading invariant, a sequence that is derived from the topology of the basins of attraction of the attracting cycle.

This paper treats the Pinsker algebra of a dynamical system in a way which avoids the use of an ordering on the acting group. This enables us to prove some of the classical results about entropy and the Pinsker algebra in the general setup of measure-preserving dynamical systems, where the acting group is a discrete countable amenable group. We prove a basic disjointness theorem which asserts the relative disjointness in the sense of Furstenberg, of $0$-entropy extensions from completely positive entropy (c.p.e.) extensions. This theorem is used to prove several classical results in the general setup. For example, we show that the Pinsker factor of a product system is equal to the product of the Pinsker factors of the component systems. Another application is to obtain a generalization (as well as a simpler proof) of the quasifactor theorem for $0$-entropy systems of Glasner and Weiss.

Flow-type suspension and homotopy suspension agree for attractor–repellor homotopy data. The connection maps associated in Conley index theory to an attractor–repellor decomposition with respect to the direct flow and its inverse are Spanier–Whitehead duals in the stably parallelizable context and are duals modulo a certain Thom construction in general.

We consider a $C^{1+1}$ diffeomorphism $f$ of a compact manifold $M$ which preserves an ergodic probability measure $\mu$. We conclude that $\mu$-a.e. $x \in M$ is contained in a disk $D_x \subset W^u(x)$, with $D_x$ open in the $W^u(x)$ topology, which exhibits an exponential volume growth rate greater than or equal to the measure-theoretic entropy of $f$ with respect to $\mu$. Drawing on results of Newhouse and Yomdin, we then find that when $f$ is $C^\infty$ and $\mu$ is a measure of maximal entropy, this exponential volume growth rate equals the topological entropy of $f$ for $\mu$-a.e. $x$.

We study the motion of a charge on a conformally flat Riemannian torus in the presence of a magnetic field. We prove that for any non-zero magnetic field there always exist orbits of this motion which have conjugate points. We conjecture that the restriction of conformal flatness of the metric is not essential for this result. This would provide a ‘twisted’ version of the recent generalization of Hopf's rigidity result obtained by Burago and Ivanov.

We prove that any expanding piecewise real-analytic map of a bounded region of the plane admits absolutely continuous invariant probability measures.

We continue the study of mixing properties of generic hyperbolic flows started in an earlier paper (D. Dolgopyat. Prevalence of rapid mixing in hyperbolic flows. Erg. Th.& Dyn. Sys.18 (1998), 1097–1114). Our main result is that generic suspension flow over subshifts of finite type is exponentially mixing. This is a quantitative version of an earlier result of Parry and Pollicott (W. Parry and M. Pollicott. Stability of mixing for toral extensions of hyperbolic systems. Proc. Steklov Inst.216 (1997), 354–363).

We study smooth local models of families of symmetric and reversible vector fields near a partially hyperbolic singularity. Special attention is given to the question of whether the involved changes of variables commute with the symmetry.

For families of conservative maps near the identity we prove the existence of open sets of parameters with persistence of homoclinic tangencies between stable and unstable leaves of ‘thick’ horse-shoes. Such families are obtained, for instance, by perturbing integrable Hamiltonian systems in $\mathbb{R}^2$ with a rapidly periodic oscillatory term and then performing a slowing change in the time variable.

The joint study of functionals of a Brownian motion $B$ and its Lévy transform $\beta= |B|-L$, where $L$ is the local time of $B$ at zero, is motivated by the conjectured ergodicity of the Lévy transform.

Here, we compute explicitly the covariance of the last zeros before time one of $B$ and $\beta$, which turns out to be strictly positive.

A minimal subshift $(X,T)$ is linearly recurrent (LR) if there exists a constant $K$ so that for each clopen set $U$ generated by a finite word, $u$, the return time to $U$, with respect to $T$, is bounded by $K|u|$. We prove that given a LR subshift $(X,T)$ the set of its non-periodic subshift factors is finite up to isomorphism. We also give a constructive characterization of these subshifts.

Equilibrium states of rational maps for Hölder continuous potentials are not $\phi$-mixing, mainly due to the presence of critical points. Here we prove that for disks the normalized return times of arbitrary orders are, in the limit, Poisson distributed as the radius of the disks go to zero. The return times are normalized by the measure of the disks. We also show that rational maps are weakly Bernoulli with respect to the partition given by Denker and Urbanski.

We study the most generic nilpotent singularity of a vector field in ${\mathbb R}^3$ which is equivariant under reflection with respect to a line, say the $z$-axis. We prove the existence of eight equivalence classes for $C^0$-equivalence, all determined by the 2-jet. We also show that in certain cases, the ${\mathbb Z}_2$-equivariant unfoldings generically contain codimension one heteroclinic cycles which are comparable to the Shil'nikov-type homoclinic cycle in non-equivariant unfoldings. The heteroclinic cycles are accompanied by infinitely many horseshoes and also have a reasonable possibility of generating suspensions of Hénon-like attractors, and even Lorenz-like attractors.

Let $\mathcal{H}$ be a hyperbolic component of quadratic rational maps possessing two distinct attracting cycles. We show that $\mathcal{H}$ has compact closure in moduli space if and only if neither attractor is a fixed point.

We consider analytically coupled circle maps (uniformly expanding and analytic) on the ${\mathbb Z}^d$-lattice with exponentially decaying interaction. We introduce Banach spaces for the infinite-dimensional system that include measures whose finite-dimensional marginals have analytic, exponentially bounded densities. Using residue calculus and ‘cluster expansion’-like techniques we define transfer operators on these Banach spaces. We get a unique (in the considered Banach spaces) probability measure that exhibits exponential decay of correlations.

This is partly exposition, partly generalization and partly completion of some of Vershik's work towards classifying sequences of measurable partitions.

We first study situations where the stable AF algebras defined by two square primitive non-singular incidence matrices with non-negative integer matrix elements are isomorphic, even though no powers of the associated automorphisms of thecorresponding dimension groups are isomorphic. More generally we consider necessary and sufficient conditions for two such matrices to determine isomorphic dimension groups.We give several examples.