**DYNAMICAL SYSTEMS Seminar**

**Thursday, June 18, 2020 (Lisbon time)**

**18 jun 2020 02:00 PM Lisboa**

Título: Lyapunov spectrum properties and continuity of the lower joint spectral radius

Orador: Reza Mohammadpour Bejargafsheh (IMPAN-Warsaw)

Abstract: In this talk we discuss ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles, and the existence of the unique equilibrium measure for subadditive potentials. We show that the restricted variational principle holds for generic cocycles over mixing subshifts of finite type, and the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show both the continuity of the entropy spectrum for such cocycles, and the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition. We also discuss the continuity of the topological pressure.

Zoom Meeting: https://videoconf-colibri.zoom.us/j/95018974726

ID da reunião: 950 1897 4726

Dispositivo móvel de um toque

+351308810988,,95018974726# Portugal

+351211202618,,95018974726# Portugal

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**DYNAMICAL SYSTEMS Seminar****Thursday, May 14, 2020, 14h (Lisbon time). **

Location: Zoom Meeting https://videoconf-colibri.zoom.us/j/96866254377

Meeting ID : 968 6625 4377

Speaker: Davi Obata (Université Cergy-Pontoise).

Title: Open sets of partially hyperbolic systems having a unique SRB measure.

Abstract: For a dynamical system, a physical measure is an ergodic invariant measure that captures the asymptotic statistical behavior of the orbits of a set with positive Lebesgue measure. A natural question in the theory is to know when such measures exist. It is expected that a \"typical\" system with enough hyperbolicity (such as partial hyperbolicity) should have such measures. A special type of physical measure is the so-called hyperbolic SRB (Sinai-Ruelle-Bowen) measure. Since the 70`s the study of SRB measures is a very active topic of research. In this talk, we will see some new examples of open sets of partially hyperbolic systems with two dimensional center having a unique SRB measure. One of the key feature for these examples is a rigidity result for a special type of measure (the so-called u-Gibbs measure) which allow us to conclude the existence of the SRB measures.

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**DYNAMICAL SYSTEMS Seminar****Thursday, May 21, 2020, 14h (Lisbon time).**

Location: Zoom Meeting https://videoconf-colibri.zoom.us/j/98242649197

Meeting ID: 982 4264 9197

Speaker: Sajjad Bakrani (Imperial College London).

Title: Invariant manifolds of homoclinic orbits: super-homoclinics and multi-pulse homoclinic loops.

Abstract: Consider a Hamiltonian flow on R4 with a hyperbolic equilibrium O and a transverse homoclinic orbit Gamma. In this talk, we discuss the dynamics near Gamma in its energy level when it leaves and enters O along strong unstable and strong stable directions, respectively. In particular, we introduce necessary and sufficient conditions for the existence of the local stable and unstable invariant manifolds of Gamma. We then consider the case in which both of these manifolds exist. We globalize them and assume they intersect transversely. We show that near any orbit of this intersection, called super-homoclinic, there exist infinitely many multi-pulse homoclinic loops.