In this paper we consider -generic families of area-preserving diffeomorphisms of the torus homotopic to the identity and their rotation sets. Let be such a family, be a fixed family of lifts and be their rotation sets, which we assume to have interior for in a certain open interval . We also assume that some rational point for a certain parameter , and we want to understand the consequences of the following hypothesis: for all , , . Under these very natural assumptions, we prove that there exists a -fixed hyperbolic saddle such that its rotation vector is . We also prove that there exists a sequence , , such that if is the continuation of with the parameter, then (the unstable manifold) has quadratic tangencies with (the stable manifold translated by ), where is any lift of to the plane. In other words, is a fixed point for , and are certain integer vectors such that do not intersect , and these tangencies become transverse as increases. We also prove that, for , has transverse intersections with , for all integer vectors , and thus one may consider that the tangencies above are associated to the birth of the heteroclinic intersections in the plane that do not exist for .

A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore it implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This, in particular, proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène’s theorem to other kinds of foliations.

Let be a closed manifold admitting a locally free action of a compact Lie group . In this paper, we study the properties of geodesic flows on given by suitable G-invariant Riemannian metrics. In particular, we will be interested in the existence of geodesics that are closed up to the action of some element in the group , since they project to closed magnetic geodesics on the quotient orbifold .

In the realm of Delone sets in locally compact, second countable Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on the autocorrelation, as well as on the density of states for random bounded operators. It is shown that for uniquely ergodic limit systems, the latter measures behave continuously with respect to the Chabauty–Fell convergence of hulls. In the special situation of Euclidean spaces, our results complement recent developments in describing spectra as topological limits: we show that the measured quantities under consideration can be approximated via periodic analogs.

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak topology) and that their entropy varies continuously with respect to the local diffeomorphism.

For any measure-preserving system with no commutativity assumptions on the transformations we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

We prove that any countable non-amenable group admits a free, minimal, amenable, purely infinite action on the non-compact Cantor set. This answers a question of Kellerhals, Monod and Rørdam [Non-supramenable groups acting on locally compact spaces. Doc. Math.18 (2013), 1597–1626].

The results of Bergelson, Host and Kra, and Leibman state that a multiple polynomial correlation sequence can be decomposed into a sum of a nilsequence (a sequence defined by evaluating a continuous function along an orbit in a nilsystem) and a null sequence (a sequence that goes to zero in density). We refine their results by proving that the null sequence goes to zero in density along polynomials evaluated at primes and along the Hardy sequence . In contrast, given a rigid sequence, we construct an example of a correlation whose null sequence does not go to zero in density along that rigid sequence. As a corollary of a lemma in the proof, the formula for the pointwise ergodic average along polynomials of primes in a nilsystem is also obtained.

We prove a conjecture by Kawaguchi–Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties.

This paper provides a systematic study of fundamental combinatorial properties of one-dimensional, two-sided infinite simple Toeplitz subshifts. Explicit formulas for the complexity function, the palindrome complexity function and the repetitivity function are proved. Moreover, a complete description of the de Bruijn graphs of the subshifts is given. Finally, the Boshernitzan condition is characterized in terms of combinatorial quantities, based on a recent result of Liu and Qu [Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift. Ann. Henri Poincaré12(1) (2011), 153–172]. Particular simple characterizations are provided for simple Toeplitz subshifts that correspond to the orbital Schreier graphs of the family of Grigorchuk’s groups, a class of subshifts that serves as the main example throughout the paper.

Let be an extension between minimal systems; we consider its relative sensitivity. We obtain that is relatively -sensitive if and only if the relative -regionally proximal relation contains a point whose coordinates are distinct; and the structure of which is relatively -sensitive but not relatively -sensitive is determined. Let be the families consisting of thick sets. We introduce notions of relative block -sensitivity and relatively strong -sensitivity. Let be an extension between minimal systems. Then the following Auslander–Yorke type dichotomy theorems are obtained: (1)  is either relatively block -sensitive or is a proximal extension where is the maximal equicontinuous factor of . (2)  is either relatively strongly -sensitive or is a proximal extension where is the maximal distal factor of .