Title: On the uniqueness of the measure of maximal entropy for geodesic flows of compact surfaces without conjugate points.

Abstract. We introduce the geodesic flow induced by metrics weaker than negative curvature. It is well-known that the geodesic flow of a compact manifold of negative curvature is Anosov. For this system we have a relatively good understanding of its dynamical and ergodic properties. However, when we weaken the metric, i.e., allowing regions of zero or positive curvature,  the system may not be Anosov and little is known about its properties. Recently, a progress was achieved that ensures that for a compact surface without conjugate points, its geodesic flow always admits a unique measure of maximal entropy that sheds some light on its ergodic theory. We sketch one of the methods used to obtain this result. 

Title: Semi rigidity of Lyapunov exponents in higher dimensions

Abstract. We present some results about partially hyperbolic diffeomorphisms f on the torus of dimension d (d > 2) which is homotopic to the linear Anosov A. In this context we show a comparison of their Lyapunov exponents, in fact, we show that the sum of the Lyapunov exponents of f is bounded by the sum of the Lyapunov exponents of the linear Anosov A. We will show this also for non-uniformly Anosov diffeomorphism with dominated decomposition, in particular for examples described by C. Bonatti and M. Viana.

This is a joint work with Ali Tahzibi.

Title: What's new in Lorenz attractors.

Abstract. Ever since its discovery in 1963 by Lorenz [1], the Lorenz attractor has been playing a central role in the research of singular flows, i.e., flows generated by smooth vector fields with singularities. In this talk I shall survey about old and new results describing the dynamics of this kind of attractors from the topological as well as the ergodic point of view. I will end sketching the proof of my newest result establishing that that in a C^1-open and densely family of vector fields (including the classical Lorenz attractor), if the point masses at singularities are not equilibrium states, then there exists a unique equilibrium state supported on Λ. In particular, there exists a unique measure of maximal entropy for the flow X|Λ. This corresponds to a joint work with Fan Yang and Jiagang Yang.

References

[1] E. N. Lorenz. Deterministic nonperiodic flow. J. Atmosph. Sci., 20:130–141, 1963.

[2] M. J. Pacifico, F. Yang and J. Yang. Entropy theory for sectional hyperbolic flows. Ann. Inst. Henri Poincar ́e, Anal. Non Lin ́eaire, 38 (2021), 1001–1030.

[3] M. J. Pacifico, F. Yang and J. Yang. Existence and uniqueness of equilibrium states for systems with specification at a fixed scale: an improved ClimenhagaThompson criterion. Nonlinearity, Volume 35, Number 12.

[4] Equilibrium states for the classical Lorenz attractor and sectional-hyperbolic attractors in higher dimensions. https://arxiv.org/abs/2209.10784

Title: Probabilistic aspects of shadowing: connection to gamblers ruin problem.

Abstract. It is well-known that shadowing holds in a neighborhood of a hyperbolic set. It is known that shadowing can hold for non hyperbolic systems, but due to results of Sakai, Abdenur, Diaz, Pilyugin, Tikhomirov shadowing is "almost" equivalent to structural stability.

At the same time numerical experiments by Hammel-Grebogi-Yorke for logistics and Henon maps shows that shadowing holds for relatively long pseudotrajectories. It poses a question which type of shadowing holds for systems, which are not necessarily hyperbolic.

I consider probabilistic approach for the topic. I show that for infinite pseudotrajectories it does not change the notion. At the same time it shows that relatively long pseudotrajectories can be shadowed by exact trajectory with high probability. The main technique is a reduction to special form of gambler's ruin problems and mild form of large deviation principle for random walks. We show that our approach works for several examples -- skew product maps.

The talk is based on joint works with G. Monakov.

Title: Transição de fase em dinâmicas suaves.

Abstract. Sabe-se que dinâmicas uniformemente expansoras ou hiperbólicas não apresentam transição de fase em relação a potenciais Holder. Em trabalhos recentes, T. Bomfim, V. Carneiro e A. Fernandes exploram a existência de transição de fase em dinâmicas suaves na vizinhança da expansividade, a fim de encontrar caracterizações de expansão neste contexto, associando com propriedades de gap espectral, análise multifractal, grandes desvios, dentre outros aspectos. Nesta palestra serão apresentados algumas das principais ideias e resultados do grupo acerca do tema. 

Title: Physical measures for non-uniform sectional expanding partially hyperbolic flows.

Abstract. We prove that a partially hyperbolic attracting set  for a $C^2$ vector field, having slow recurrence to equilibria, supports an ergodic physical measure if, and only if, the trapping region admits non-uniform sectional expansion on a positive  Lebesgue measure subset. Moreover, in this case, the attracting set supports at most finitely many ergodic physical measures.

This extends to continuous time systems, a similar well-known result obtained for diffeomorphisms, encompassing the presence of equilibria accumulated by regular orbits within the attracting set.