Author: Yi Pan

Mauricio Poletti (Universidade Federal do Ceará)

For $C^(l+)$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. In this talk I will give two the applications: Uniqueness of MME for finite horizon dispersing billiards and the robustly non-uniformly hyperbolic volume-preserving endomorphisms introduced by Andersson-Carrasco-Saghin.

Yunzhe Li (ISTA)

We construct a nontrivial analytic family of standard maps in which the Lyapunov exponents of infinitely many periodic orbits remain constant throughout the family. After a brief review of the background and related results, I will discuss key components of the proof, including a resonant normal form for discrete maps and an iterated correction mechanism.

Sébastien Biebler (Université Paris Cité)

One of the main goals in the theory of dynamical systems is to describe the dynamics of a “typical” map. For instance, in the case of diffeomorphisms of a given manifold, it was conjectured by Smale in the 60s that uniform hyperbolicity was generically satisfied. This hope was however fast discouraged by exhibiting dynamical systems displaying in a robust way dynamical configurations which are obstructions to hyperbolicity: robust homoclinic tangencies (this is the so-called Newhouse phenomenon) and robust heterodimensional cycles. In this talk, I will explain these phenomena and their extensions to the complex setting. In particular, I will show how to construct robust heterodimensional cycles in the family of polynomial automorphisms of C³. The main tool is the notion of blender coming from real dynamics.

Jérôme Buzzi (CNRS, Laboratoire de Mathématiques d’Orsay)

Santiago Martinchinch introduced Discretized Anosov flows, a large class of partially hyperbolic diffeomorphisms that includes deformations of time-one maps of transitive Anosov flows. Under an assumption of irreducibility, we establish the following dichotomy

– either there are exactly two ergodic measures maximizing the entropy (or MMEs);
 – or there is a unique MME with zero center exponent.
The hyperbolic case is open and dense and implies that the MMEs are Bernoulli and exponentially mixing (they satisfy Strong Positive Recurrence).
This is a joint work in progress with Sylvain CROVISIER, Mauricio POLETTI, and Ali TAHZIBI.

Michael Yampolsky (University of Toronto)

I will give a survey of the renormalization picture for rotational dynamics which is emerging in one and several dimensions, and will discuss some consequences, such as the smoothness of Arnold tongues and the existence of rotation domains for multi-dimensional maps. 

Otto Vaughn Osterman (University of Maryland)

We consider the planar circular restricted three-body problem, modeling the motion of a massless asteroid in the plane undergoing gravitational attraction toward two bodies, each of which moves in a circular path around their common center of mass. For small mass ratios, the motion of the asteroid is approximated by the Kepler problem when the asteroid is far from a collision, and a large set of Kepler motions in which the paths of the asteroid and the smaller body do not intersect persist as quasi-periodic motions in the perturbed system. However, these quasi-periodic motions with incommensurable frequencies are not possible for Kepler motions in which the paths intersect due to the potential for close interactions between the asteroid and the smaller body. The existence of hyperbolic sets in which the asteroid repeatedly comes close to a collision was proven independently by Bolotin and MacKay and by Font, Nunes, and Simó. My result, currently in preparation, is that there exists stable motions of the asteroid near resonant Kepler orbits in which the asteroid repeatedly undergoes close interactions with the smaller body.

Xianghui Shi (Peking University)

Expanding Thurston maps were introduced by M. Bonk and D. Meyer with motivation from complex dynamics and Cannon’s conjecture from geometric group theory via Sullivan’s dictionary. In this talk, we explore the ergodic theory surrounding these maps and present some recent advancements. We demonstrate that the entropy map is upper semi-continuous if and only if the map has no periodic critical points. Furthermore, we show that ergodic measures are entropy-dense and derive level-2 large deviation principles for Birkhoff averages, periodic points, and iterated preimages.    The main tools used in the proof are called subsystems of expanding Thurston maps, which naturally emerge in the study of dynamics on subsets. We develop the thermodynamic formalism for subsystems and establish the existence, uniqueness, and ergodic properties of equilibrium states for Hölder continuous potentials. This is joint work with Zhiqiang Li.

Malo Jézéquel (CNRS, LMBA-Brest)

To an Anosov flow (i.e. a smooth uniformly hyperbolic flow on a closed manifold), one may associate a zeta function. This is a meromorphic function defined in term of the periodic orbits of the flow, that can be used for instance to count these orbits. After recalling this definition, I will explain how to extend it to certain “smooth pseudo-Anosov flows” on 3-manifolds, a class of flows that looks like Anosov flows except for a finite number of singular orbits. The motivation for studying zeta functions for these flows comes from topology. In particular, I will discuss a version of Fried’s conjecture for smooth pseudo-Anosov flows. This is a joint work with Jonathan Zung.

Silvius Klein (Pontifical Catholic University of Rio de Janeiro)

In this talk I will discuss some recent results on large deviations type estimates for certain skew-product dynamical systems. Moreover, I will describe an abstract approach for deriving such statistical properties, based on the strong mixing properties of an appropriate Markov operator.[Based on joint work with Ao Cai (Soochow University) and Pedro Duarte (University of Lisbon).]

Stefano Marmi  (Scuola Normale Superiore di Pisa)

Addressing a question raised by Kolmogorov and Herman, I will review previous work on the analytic structure in the complex frequency plane of solutions of certain small divisor problems and its relation with with the notion of quasianalytic function of a complex variable. I will discuss both the linear and nonlinear case.Most of the talk will be based on papers with C. Carminati, D. Sauzin and A. Sorrentino