On the space of signed invariant measures of Aℕ, one constructs a norm (and hence a distance) that seems to have a particular significance in dynamics. I shall present some of its properties, in particular a duality theorem à la Kantorovich–Rubinshtein, which gives an expression of this distance using couplings.

We prove the existence of a topologically mixing interval exchange transformation and prove that no interval exchange is topologically mixing of all orders.

We study the limiting behavior of multiple ergodic averages involving several, not necessarily commuting, measure-preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and another that uses iterates along shifted polynomials. We prove pointwise convergence in both cases, thus answering a question of I. Assani in the former case, and extending the results of B. Host and B. Kra, and A. Leibman in the latter case. Our argument is based on some elementary uniformity estimates of general bounded sequences, decomposition results in ergodic theory, and equidistribution results on nilmanifolds.

We study deterministic and stochastic perturbations of incompressible flows on a two-dimensional torus. Even in the case of purely deterministic perturbations, the long-time behavior of such flows can be stochastic. The stochasticity is caused by instabilities near the saddle points as well as by the ergodic component of the locally Hamiltonian system on the torus.

If a countable amenable group G contains an infinite subgroup Γ, one may define, from a measurable action of Γ, the so-called co-induced measurable action of G. These actions were defined and studied by Dooley, Golodets, Rudolph and Sinelsh’chikov. In this paper, starting from a topological action of Γ, we define the co-induced topological action of G. We establish a number of properties of this construction, notably, that the G-action has the topological entropy of the Γ-action and has uniformly positive entropy (completely positive entropy, respectively) if and only if the Γ-action has uniformly positive entropy (completely positive entropy, respectively). We also study the Pinsker algebra of the co-induced action.

In the first part of this paper, we consider several natural problems about locally homogeneous rigid geometric structures. In particular, we formulate a notion of topological completeness which is adapted to the study of global rigidity of chaotic dynamical systems. In the second part of the paper, we prove the following result: let φ be a C∞ expanding map of a closed manifold. If φ preserves a topologically complete C∞ rigid geometric structure, then φ is C∞ conjugate to an expanding infra-nilendomorphism.

In this paper we study the asymptotic behaviour of the escape rate of a Gibbs measure supported on a conformal repeller through a small hole. There are additional applications to the convergence of the Hausdorff dimension of the survivor set.

In this paper, we introduce and study the α-Farey map and its associated jump transformation, the α-Lüroth map, for an arbitrary countable partition α of the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called α-sum-level sets for the α-Lüroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the α-Farey map and the α-Lüroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partition α.

We consider N-body problems with potential 1/r2κ, where κ∈(0,1), including the Newtonian case (κ=1/2). Given R>0 and T>0, we find a uniform upper bound for the minimal action of paths binding, in time T, any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates the Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding two configurations in free time). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax–Oleinik semigroup, and we show that they are global viscosity solutions of the corresponding Hamilton–Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

Let Γ be a topological semigroup acting on a topological space X, and let Γ0 be a subsemigroup of Γ. We give general conditions ensuring that Γ and Γ0 have the same transitive points.

The complexity function of an infinite word w on a finite alphabet A is the sequence counting, for each non-negative integer n, the number of words of length n on the alphabet A that are factors of the infinite word w. Let f be a given function with subexponential growth. The goal of this work is to estimate the generalized Hausdorff dimensions of the set of real numbers whose q-adic expansion has a complexity function bounded by f and the set of real numbers whose continued fraction expansion is bounded by q and has a complexity function bounded by f.

In the paper, we prove convergence of moments of all orders for Axiom A diffeomorphisms and flows. The same results hold for non-uniformly hyperbolic diffeomorphisms and flows modelled by Young towers with superpolynomial tails. For polynomial tails, we prove convergence of moments up to a certain order, and give examples where moments diverge when this order is exceeded. Non-uniformly hyperbolic systems covered by our result include Hénon-like attractors, Lorenz attractors, semidispersing billiards, finite horizon planar periodic Lorentz gases, and Pomeau–Manneville intermittency maps.

In this paper, we focus on the packing measures of self-similar sets. Let K be a self-similar set whose Hausdorff dimension and packing dimension equal s. We state that if K satisfies the strong open set condition with an open set 𝒪, then for each open ball B(x,r)⊂𝒪 centred in K, where 𝒫s denotes the s-dimensional packing measure. We use this inequality to obtain some precise density theorems for the packing measures of self-similar sets. These theorems can be used to compute the exact value of the s-dimensional packing measure 𝒫s (K) of K. Moreover, by using the above results, we show the continuity of the packing measure function of the attractors on the space of self-similar iterated function systems satisfying the strong separation condition. This result gives a complete answer to a question posed by Olsen in [15].

We study the set of ergodic measures for a Markov semigroup on a Polish state space. The principal assumption on this semigroup is the e-property, an equicontinuity condition. We introduce a weak concentrating condition around a compact set K and show that this condition has several implications on the set of ergodic measures, one of them being the existence of a Borel subset K0 of K with a bijective map from K0 to the ergodic measures, by sending a point in K0 to the weak limit of the Cesàro averages of the Dirac measure on this point. We also give sufficient conditions for the set of ergodic measures to be countable and finite. Finally, we give a quite general condition under which the Cesàro averages of any measure converge to an invariant measure.

In this paper, we first discuss some connections between template theory and the description of basic sets of Smale flows on 3-manifolds due to F. Béguin and C. Bonatti. The main tools we use are symbolic dynamics, template moves and some combinatorial surgeries. Secondly, we obtain some relationship between the surgeries and the number of S1×S2 factors of M for a non-singular Smale flow on a given closed orientable 3-manifold M. We also prove that any template T can model a basic set Λ of a non-singular Smale flow on nS1×S2 for some positive integer n.