We prove that for a generic sub-Riemannian and reversible sub-Finsler metrics defined on a fixed co-rank 1 distribution, all strictly normal periodic orbits are non-degenerate.

of Maryland in 2008 to recognize mathematicians who have made substantial impact in dynamical systems

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We discuss a remarkable phenomenon which occurs in one-dimensional dynamics, that is, for maps acting on an interval or a circle. Such maps are often not structurally stable, however their topological class is an infinite dimensional smooth manifold with finite codimension. Consequently, the deformation theory of such systems is quite rich. Recent developments suggest that the study of the existence, uniqueness and regularity of solutions for certain cohomological equations is crucial for deeper understanding of these phenomena, and ergodic theory plays an important role in this study.

Let X = G/Γ, where G is a Lie group and Γ is a lattice in G, let O be an open subset of X, and let F = {gt : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F -orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Ad-diagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms.

The mathematical activity of Tim Austin has been, since the very beginning, quite abundant and versatile. We will describe and comment on three of his results which were selected as most representative for the Brin Prize Award and which culminated in the proof of the weak Pinsker structure theorem.

For any accessible partially hyperbolic homogeneous flow, we show that all smooth time changes are K and hence mixing of all orders. We also establish stable ergodicity for time-one map of these time changes.

On the space L n+1 of unimodular lattices in R n+1 , we consider the standard action of a(t) = diag(t n , t −1 , . . . , t −1 ) ∈ SL(n + 1, R) for t > 1. Let M be a nondegenerate submanifold of an expanding horospherical leaf in L n+1 . We prove that for all x ∈ M \ E and t > 1, if µ x,t denotes the normalized Lebesgue measure on the ball of radius t −1 around x in M , then the translated measure a(t)µ x,t gets equidistributed in L n+1 as t → ∞, where E is a union of countably many lower dimensional submanifolds of M . In particular, if µ is an absolutely continuous probability measure on M , then a(t)µ gets equidistributed in L n+1 as t → ∞. This result implies the non-improvability of Dirichlet's Diophantine approximation theorem for almost every point on a C n+1 -submanifold of R n satisfying a non-degeneracy condition, answering a question arising from the work of Davenport and Schmidt (1969).

The goal of this article is to establish several general properties of a somewhat large class of partially hyperbolic diffeomorphisms called discretized Anosov flows. A general definition for these systems is presented and is proven to be equivalent with the definition introduced in [BFFP19], as well as with the notion of flow type partially hyperbolic diffeomorphisms introduced in [BFT20]. The set of discretized Anosov flows is shown to be C 1 -open and closed inside the set of partially hyperbolic diffeomorphisms. Every discretized Anosov flow is proven to be dynamically coherent and plaque expansive. Unique integrability of the center bundle is shown to happen for whole connected components, notably the ones containing the time 1 map of an Anosov flow. For general connected components, a result on uniqueness of invariant foliation is obtained. Similar results are seen to happen for partially hyperbolic systems admitting a uniformly compact center foliation extending the studies initiated in [BB16].