We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of ‘symmetric measure’: exchangeability and the Gibbs (or conformal) property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one-dimensional case) exhibit a variety of spectral properties.

It is shown how regular model sets can be characterized in terms of the regularity properties of their associated dynamical systems. The proof proceeds in two steps. First, we characterize regular model sets in terms of a certain map $\beta$ and then relate the properties of $\beta$ to those of the underlying dynamical system. As a by-product, we can show that regular model sets are, in a suitable sense, as close to periodic sets as possible among repetitive aperiodic sets.

We study frequently hypercyclic operators, a natural new concept in hypercyclicity that was recently introduced by F. Bayart and S. Grivaux. We derive a strengthened version of their Frequent Hypercyclicity Criterion, which allows us to obtain examples of frequently hypercyclic operators in a straightforward way. Moreover, Bayart and Grivaux have noted that the frequent hypercyclicity setting differs from general hypercyclicity in that the set of frequently hypercyclic vectors need not be residual. We show here that, under weak assumptions, this set is only of first category. Motivated by this we study the question of whether one may write every vector in the underlying space as the sum of two frequently hypercyclic vectors. This investigation leads us to the introduction of a new notion, that of Runge transitivity.

We discuss the asymptotic distribution of the directions in homology of periodic orbits of Anosov flows. We obtain a limiting measure which is either a Dirac measure on a single point or is fully supported. In the latter case, we relate the result to a more general equidistribution problem.

We study the ergodic properties of the additive Euclidean algorithm $f$ defined in $\mathbb{R}^2_+$. A natural extension of $f$ is obtained using the action of ${\it SL}(2, \mathbb{Z})$ on a subset of ${\it SL}(2, \mathbb{R})$. We prove that, while $f$ is an ergodic transformation with an infinite invariant measure equivalent to the Lebesgue measure, the invariant measure is not unique up to scalar multiples, and in fact there is a continuous family of such measures.

We study the problem of conservation of maximal and lower-dimensional invariant tori for analytic convex quasi-integrable Hamiltonian systems. In the absence of perturbation the lower-dimensional tori are degenerate, in the sense that the normal frequencies vanish, so that the tori are neither elliptic nor hyperbolic. We show that if the perturbation parameter is small enough, for a large measure subset of any resonant submanifold of the action variable space, under some generic non-degeneracy conditions on the perturbation function, there are lower-dimensional tori which are conserved. They are characterized by rotation vectors satisfying some generalized Bryuno conditions involving also the normal frequencies. We also show that, again under some generic assumptions on the perturbation, any torus with fixed rotation vector satisfying the Bryuno condition is conserved for most values of the perturbation parameter in an interval small enough around the origin. According to the sign of the normal frequencies and of the perturbation parameter the torus becomes either hyperbolic or elliptic or of mixed type.

We prove that a ‘positive probability’ subset of the boundary of ‘{uniformly expanding circle transformations}’ consists of Kupka–Smale maps. More precisely, we construct an open class of two-parameter families of circle maps $(f_{a,\theta})_{a,\theta}$ such that, for a positive Lebesgue measure subset of values of $a$, the family $(f_{a,\theta})_\theta$ crosses the boundary of the uniformly expanding domain at a map for which all periodic points are hyperbolic (expanding) and no critical point is pre-periodic. Furthermore, these maps admit an absolutely continuous invariant measure. We also provide information about the geometry of the boundary of the set of hyperbolic maps.

The existence of non-continuous invariant graphs (or strange non-chaotic attractors) in quasiperiodically forced systems has generated great interest, but there are still very few rigorous results about the properties of these objects. In particular, it is not known whether the topological closure of such graphs is typically a filled-in set, i.e. consists of a single interval on every fibre, or not. We give a positive answer to this question for the class of so-called pinched skew products, where non-continuous invariant graphs occur generically, provided that the rotation number on the base is diophantine and the system satisfies some additional conditions. For typical parameter families these conditions translate to a lower bound on the parameter. On the other hand, we also construct examples where the non-continuous invariant graphs contain isolated points, such that their topological closure cannot be filled-in.

We investigate the convergence, in norm and almost everywhere (a.e.), of weighted ergodic averages as well as weighted sums of independent identically distributed (iid) random variables. The averages are true ones, normalized by the corresponding sums of weights, which are only assumed to be non-negative. The $L_2$-norm convergence in the mixing case is shown to rely upon very simple conditions on the weights. We show that ‘quasimonotone weights’ with a simple additional condition yield a.e. convergence of weighted averages for all Dunford–Schwartz contractions of probability spaces and $L_1$-functions. For independent random variables, we look at weighted averages of centered random variables with bounded variances (or bounded moments of some order greater than 1), in particular the iid case, and obtain several sufficient conditions on the weights for almost sure convergence (weighted SLLN). For example, in Theorem 4.14 we show that if a weight sequence $\{w_k\}$ with divergent partial sums $W_n$ satisfies

\[ \sup_{n\ge 1} \frac1{W_n} \sum_{k=1}^n w_k ({\rm log}(w_k+1))^\beta < \infty\quad \text{for some }\beta > 1 \]

then for any iid sequence in the class $L({\rm log}^+L)^{1+\epsilon}$ the weighted averages converge almost surely to the expectation.

We show relations between new notions on cellular automata based on probabilistic dynamics: almost everywhere sensitivity to initial conditions for Besicovitch pseudo-distance, damage spreading (which measures the information (or damage) propagation), and the destruction of the initial configuration information. Through natural examples, we illustrate the relations between these formal definitions.

An algebraic $\mathbb{Z}^d$-action of entropy rank one is one for which each element has finite entropy. Using the structure theory of these actions due to Einsiedler and Lind, this paper investigates dynamical zeta functions for elements of the action. An explicit periodic point formula is obtained leading to a uniform parameterization of the zeta functions that arise in expansive components of an expansive action, together with necessary and sufficient conditions for rationality in a more general setting.

Let $X$ be a compact space, $f\colon X \to X$ a continuous map, and $\Lambda \subset X$ be any $f$-invariant subset. Assume that there exists a nested family of subsets $\{\Lambda_l\}_{l \geq 1}$ that exhaust $\Lambda$, that is $\Lambda_l \subset\Lambda_{l+1}$ and $\Lambda =\bigcup_{l \geq 1} \Lambda_l$. Assume that the potential $\varphi \colon X \to \mathbb{R}$ is continuous on the closure of each $\Lambda_l$ but not necessarily continuous on $\Lambda$. We define the topological pressure of $\varphi$ on $\Lambda$. This definition is shown to have a corresponding variational principle. We apply the topological pressure and variational principle to systems with non-zero Lyapunov exponents, countable Markov shifts, and unimodal maps.

We introduce and study a non-uniform hyperbolicity condition for complex rational maps that does not involve a growth condition. We call this condition backward contraction. We show this condition is weaker than the Collet–Eckmann condition, and than the summability condition with exponent one. Our main result is a connecting lemma for backward-contracting rational maps, roughly saying that we can perturb a rational map to connect each critical orbit in the Julia set with an orbit that does not accumulate on critical points. The proof of this result is based on Thurston's algorithm and some rigidity properties of quasi-conformal maps. We also prove that the Lebesgue measure of the Julia set of a backward-contracting rational map is zero, when it is not the whole Riemann sphere. The basic tool of this article is sets having a Markov property for backward iterates that are holomorphic analogues of nice intervals in real one-dimensional dynamics.