Résumé : The GIKN construction was introduced by Gorodetski, Ilyashenko, Kleptsyn, and Nalsky. It gives a nonhyperbolic ergodic measure which is a weak-* limit of a special sequence of measures supported on periodic orbits.
This method was later adapted by numerous authors (Bonatti, Cheng, Crovisier, Diaz, Gan, Wang, Yang, Zhang) and provided examples of nonhyperbolic invariant measures in various settings.
We prove that the result of the GIKN construction is always a loosely Kronecker measure in the sense of Ornstein, Rudolph, and Weiss (equivalently, standard measure in the sense of Katok, another name is loosely Bernoulli measure with zero entropy). For a proof we introduce and study the Feldman-Katok pseudometric F. The pseudodistance F is a topological counterpart of the f-bar metric for finite-state stationary stochastic processes introduced by Feldman and, independently, by Katok, later developed by Ornstein, Rudolph, and Weiss. We show that every measure given by the GIKN construction is the F-limit of a sequence of periodic measures. On the other hand we prove that a measure which is the F-limit of a sequence of ergodic measures is ergodic and its entropy is smaller or equal than the lower limit of entropies of measures in the sequence. Furthermore we demonstrate that F-Cauchy sequence of periodic measures tends in the weak-* topology to a loosely Kronecker measure. The talk will be based on a joint work with Dominik Kwietniak.
Lieu : Salle 318
Résumé : Non-self-adjoint operators appear in many settings, from kinetic theory and quantum mechanics to linearizations of equations of mathematical physics. The spectral analysis of such operators, while often notoriously difficult, reveals a wealth of new phenomena, compared with their self-adjoint counterparts. Spectra for non-self-adjoint operators display fascinating features, such as lattices of eigenvalues for operators of Kramers-Fokker-Planck type, say, and eigenvalues for operators with analytic coefficients in dimension one, concentrated to unions of curves in the complex spectral plane. In this talk, we shall discuss spectra for non-self-adjoint perturbations of self-adjoint operators in dimension two, under the assumption that the classical flow of the unperturbed part is completely integrable. We shall describe the role played by the flow-invariant Lagrangian tori of the completely integrable system, both Diophantine and rational, in the spectral analysis of the non-self-adjoint operators. The particular focus will be on the recent results on spectral contributions of rational tori, leading to eigenvalues having the form of the "legs in a spectral centipede". This talk is based on joint work
with Johannes Sjöstrand.