In this paper we obtain (almost) optimal results concerning randomly weighted one-sided ergodic Hilbert transforms. Given an iid sequence of centered random variables (Xn) in L log L, we show that there exists a universal set $\Omega'$ of probability 1 such that for any $\omega\in \Omega'$ the realization $(X_n(\omega))$ is good for the almost everywhere convergence of the weighted one-sided ergodic Hilbert transform associated with any dynamical system and any $g\in L\log L$. The method applies to powers along subsequences with ‘small’ growth and when considering Dunford–Schwartz operators instead of pointwise transformation. If the (Xn) are symmetric, but only in L log log L, we obtain a slightly weaker result.

We investigate the relation between preimage multiplicity and topological entropy for continuous maps. An argument originated by Misiurewicz and Przytycki shows that if every regular value of a C1 map has at least m preimages then the topological entropy of the map is at least log m. For every integer, there exist continuous maps of the circle with entropy 0 for which every point has at least m preimages. We show that if in addition there is a positive uniform lower bound on the diameter of all pointwise preimage sets, then the entropy is at least log m.

A result of Krengel on the density of the set of vectors with infinitely many pairwise orthogonal iterates under powers of a unitary operator with continuous spectrum is generalized to actions of countable Abelian groups.

In this paper we consider the family of rational maps of the complex plane given by \[z^2+\frac{\lambda}{z^2}\] where $\lambda$ is a complex parameter. We regard this family as a singular perturbation of the simple function $z^2$. We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the corresponding maps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However, we also show that parameters corresponding to different open sets have dynamics that are not conjugate.

A block code from an irreducible shift of finite type can be lifted canonically through resolving maps to a resolving map. There is an application to Markovian maps.

We prove structure theorems for holomorphic correspondences realizing matings between pinched polynomial-like maps with connected Julia sets and Hecke groups (Fuchsian representations of free products C2 * Cn of cyclic groups of orders 2 and n). We show that such matings are generated by two-to-two subcorrespondences if and only if the maps are Chebyshev-like. We describe the dynamical behaviour of these matings in detail in the case n = 4 and we present a conjectured combinatorial description of the connectedness locus in parameter space in this case.

A graph surface P is a two-dimensional polyhedron having the simplest kind of non-trivial singularities which result from gluing surfaces with compact boundaries along boundary components. We study the behavior of the volume entropy h(g) of hyperbolic metrics g on a closed graph surface P depending on the lengths of singular geodesics $Q\subset P$. We show that always h(g) > 1 and $h(g)\to\infty$ as $L_g(Q)\to\infty$ for at least one singular geodesic Q.

I would like to mention here several improvements or additional results to the paper ‘On randomly weighted one-sided ergodic Hilbert transforms’. We obtained there that iid random weights in L log L are universally good for the almost sure convergence of the weighted one-sided ergodic Hilbert transform for any Dunford–Schwartz operator in a probability space $(X,\Sigma, \mu)$ and any function g satisfying $\int_X |g|{\rm log}^+|g| \,d\mu<+\infty$.

We show that the set of cocycles with integral separateness is open and dense in the space of all bounded $Gl(d,{\mathbb R})$-cocycles equipped with uniform topology. As a consequence, a generic bounded linear cocycle has simple Lyapunov spectrum and dominated Oseledets splitting, and a generic bounded $Sl(2,\mathbb R)$-cocycle is uniformly hyperbolic.

This work is devoted to the study of periodic solutions of generalized Lotka–Volterra systems in three dimensions. The group A3 of isometries of the regular tetrahedron acts on the twelve-dimensional space of such systems. We find four components (modulo the action of A3) of the center variety, i.e. the semi-algebraic subset consisting of systems with a family of periodic solutions (which degenerate at the center). We focus attention on the component which consists of systems X0 with an invariant plane L0 transversal to the coordinate axes and with a (partial) first integral on L0. Under the assumption of normal hyperbolicity of X0 along L0 we begin study of limit cycles appearing for a perturbed system $X_{0}+\varepsilon X_{1}$ (in the Lotka–Volterra class). The linearization of the problem leads to the problem of zeroes of certain Poincaré–Melnikov integrals of a new type. We determine the asymptotic behavior of these integrals near two critical values of the partial first integral of the restricted system X0|L0. We also study the situation near the intersection of different components of the center variety.

We define and study new invariants called pre-image entropies which are similar to the standard notions of topological and measure-theoretic entropies. These new invariants are only non-zero for non-invertible maps, and they give a quantitative measurement of how far a given map is from being invertible. We obtain analogs of many known results for topological and measure-theoretic entropies. In particular, we obtain product rules, power rules, analogs of the Shannon–Breiman–McMillan theorem, and a variational principle.

For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost-sure central limit theorem (ASCLT). In fact, we provide a rate of convergence in the Kantorovich distance. Maxima of partial sums are also shown to obey an ASCLT. The key tool is an exponential inequality recently obtained. Then we establish (optimal) almost-sure convergence rates for the supremum of moving averages of Lipschitz observables (Erdös–Rényi-type law). This is done by refining the usual large-deviations estimates available for expanding maps of the interval. We end up with an application to entropy estimation ASCLTs that refine the Shannon–McMillan–Breiman and Ornstein–Weiss theorems.

The geometry of Cantor systems associated with an asymptotically non-hyperbolic family $(f_\epsilon)_{0\le\epsilon\le \epsilon_0}$ was studied by Jiang (Geometry of Cantor systems. Trans. Amer. Math. Soc.351 (1999), 1975–1987). By applying the geometry studied there, we prove that the Hausdorff dimension of the maximal invariant set of $f_\epsilon$ behaves like $1- K\epsilon^{1/\gamma}$ asymptotically, as was conjectured by Jiang (Generalized Ulam–von Neumann transformations. PhD Thesis, CUNY Graduate Center, May 1999).

We define the concept of horospherical points for flows admitting a local product structure, and show that these points have a dense strong stable set. This generalizes a classical result from G. A. Hedlund, in the context of geodesic flows on negatively curved manifolds.

We characterize when a lower entropy locally compact subshift S embeds into a locally compact mixing Markov shift T. As in the compact case (when T is a shift of finite type) the existence of an embedding depends on S only through its periodic orbit counts. However, in contrast to the compact case, the topology of the target system T becomes important. This is demonstrated with the help of the Zeta Function Lemma, which in particular characterizes the periodic orbit counts and entropies of locally compact mixing Markov shifts.

We realize the Hecke C*-algebra $\mathcal{C}_{\mathbb{Q}}$ of Bost and Connes as a direct limit of Hecke C*-algebras which are semigroup crossed products by $\mathbb{N}^F$, for F a finite set of primes. For each approximating Hecke C*-algebra we describe a composition series of ideals. In all cases there is a large type I ideal and a commutative quotient, and the intermediate subquotients are direct sums of simple C*-algebras. We can describe the simple summands as ordinary crossed products by actions of $\mathbb{Z}^S$ for S a finite set of primes. When $\vert S\vert =1$, these actions are odometers and the crossed products are Bunce–Deddens algebras; when $\vert S\vert >1$, the actions are an apparently new class of higher-rank odometer actions, and the crossed products are an apparently new class of classifiable AT-algebras.

We consider families of one-dimensional maps on the circle which mix at sub-polynomial rates. Such maps will have an indifferent fixed point, and we show that the rate of mixing of these maps is determined by the precise degeneracy of the fixed point. By constructing suitable induced maps with a countable Markov partition and applying a probabilistic coupling argument we are able to give good estimates on the rate of mixing for these families of maps. The underlying question we pose is the following: given any specific rate of mixing can we construct a map which has that rate of mixing? For a large class of specific rates we explain how to construct the appropriate map.

We consider zero-noise limits of random perturbations of dynamical systems and examine, in terms of the continuity of entropy and Lyapunov exponents, circumstances under which these limits are Sinai–Ruelle–Bowen (SRB) measures. The ideas of this general discussion are then applied to specific classes of attractors. We prove, for example, that partially hyperbolic attractors with one-dimensional center subbundles always admit SRB measures.

For locally compact groups all actions on a standard measure algebra have a spatial realization. For many Polish groups this is no longer the case. However, we show here that for non-archimedean Polish groups all measure algebra actions do have spatial realizations. In the other direction we show that an action of a Polish group is whirly (‘ergodic at the identity’) if and only if it admits no spatial factors and that all actions of a Lévy group are whirly. We also show that in the Polish group $\rm{Aut\,}(X,\mathcal{X},\mu)$, for the generic automorphism T the action of the subgroup $\Lambda(T)=\rm{cls\,}\{T^n: n\in \mathbb{Z}\}$ on the Lebesgue space $(X,\mathcal{X},\mu)$ is whirly.

Measure equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field has grown considerably over the past few years, developing tools to distinguish between different ME classes of countable groups. On the other hand, constructions of ME groups are very rare. We present a new method, based on a notion of measurable free factor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about ME for countable groups.