24º SIES: 30 de outubro de 2025 (quinta-feira)

The emergence of a dynamical system is a quantitative measure of how far the system is from being ergodic. In this seminar, we study emergence for a class of paradigmatic examples: flows whose every orbit is periodic but whose period function is unbounded. After recalling the definition of emergence, we provide a criterion for detecting low emergence and show that this criterion applies to several classical constructions of such flows. Finally, we present a construction of a counterexample to the Periodic Orbit Conjecture with positive order of emergence.

The Pugh-Shub conjecture states that, most volume-preserving partially hyperbolic systems are ergodic. The most successful program to prove this conjecture involves dividing the conjecture into two subconjectures (1) most partially hyperbolic systems are accessible; and (2) that any accessible volume-preserving system is ergodic. These conjectures have inspired a large body of research for diffeomorphisms which can be adapted to the case of non-invertible dynamical systems. The best result in the direction of Conjecture 1 is by Burns and Wilkinson, who established the conjecture for C²-diffeomorphisms under an additional hypothesis known as center bunching. Our result generalises this work for endomorphisms leaving only the center bunching hypothesis used in the original Burns and Wilkinson work and an additional constant Jacobian determinant hypothesis. Together with He’s result from 2017, our work proves the conjectures in the context of non-invertible systems, assuming that the center is one-dimensional and the endomorphism has a constant Jacobian determinant.

We build symbolic dynamics for non-uniformly hyperbolic flows in any dimension. This work extends the construction made by Buzzi, Crovisier and Lima in dimension 3. This is a work in progress, in collaboration with Yuri Lima and Juan Mongez.

We give sufficient conditions such that partially hyperbolic diffeomorphisms that admit a more general version of the notion Some Hyperbolicity, so called SH saddle property introduced by Piñeyrúa in [P23], are robustly topologically transitive. Furthermore, we present a new large class of examples that are partially hyperbolic skew product maps in T4 with non trivial, non hyperbolic action on the homology, and one of the fiber is the diffeomorphism topologically conjugate to the Thurston's Pseudo-Anosov map obtained in [GK82]. We show that this map satisfies the hypotheses of the main result. In particular, the two dimensional tangent bundle of the fiber neither admits any one-dimensional invariant direction, nor does it have a non-uniform expanding/contracting behavior.
References:
[P23] Piñeyrúa, Luis Pedro. "Some hyperbolicity revisited and robust transitivity." (2023)., arXiv:2302.01914.
[GK82] Gerber, Marlies and Anatole Katok. "Smooth models of Thurston's pseudo-Anosov maps." Annales Scientifiques De L Ecole Normale Superieure 15 (1982): 173-204.

We extend D. Burguet’s construction of SRB measures [1] for the non-invertible scenario obtaining hyperbolic invariant measures with absolutely continuous disintegrations on stable manifolds for a certain class of endomorphisms on the two torus. The constructed measures maximize the folding entropy, in particular, one may obtain such SRB measures for conservative perturbations of the examples given by M. Andersson, P. Carrasco and R. Saghin [2] for which the Lebesgue measure does not maximize the folding entropy. This way, we obtain examples of topologically mixing maps with at least two inverse SRB measures. In the case of inverse SRB measures that maximize the folding entropy, we give criteria for uniqueness.
References:
[1] D. Burguet, SRB measures for C∞ surface diffeomorphisms, Invent. Math. 235, 1019-1062 (2024).
[2] M. Andersson, P. Carrasco, R. Saghin, Non-uniformly hyperbolic endomorphisms, to appear in Compositio Mathematicae (2025)