**DYNAMICAL SYSTEMS Webinar**

**Isospectral Reductions, Attractors and Spectral Equivalence**

**Thursday, October 22, 2020 (Lisbon time)**

**Speaker: Longmei Shu **(Emory University).

**Abstract:** Isospectral transformations (IT) of matrices and networks allow for compression of either object, a matrix or network, while keeping all the information about their eigenvalues and eigenvectors. We analyze what happens to the generalized eigenvectors under isospectral transformations. Isospectral compressions are performed with respect to some chosen characteristic of nodes or edges of the network. Each isospectral compression defines a dynamical system on the space of all networks. We show that any orbit of such a dynamical system with a finite network as the initial point converges to an attractor. Such an attractor is a smaller network where a chosen characteristic has the same value for all nodes or edges. We demonstrate that isospectral contractions of one and the same network defined by different characteristics may converge to the same as well as to different attractors. We also show that networks that are spectrally equivalent with respect to some characteristic could be non-spectrally equivalent for another characteristic

**Zoom Meeting: LxDS webinar****Time: Oct 22, 2020 02:00 PM Lisbon **

Join from PC, Mac, Linux, iOS or Android: https://videoconf-colibri.zoom.us/j/87278591614

Or iPhone one-tap: 308810988,87278591614# or 211202618,87278591614#

Or Telephone:

Dial: +351 308 810 988 (Portugal Toll) or +351 211 202 618 (Portugal Toll)

Meeting ID: 872 7859 1614

https://videoconf-colibri.zoom.us/u/kevcZaZwCS

**New trends in Lyapunov exponents**

Time: **12:45 PM Lisbon** Date: **7th of July, 2020**

Zoom Meeting: https://puc-rio.zoom.us/j/91404010846?pwd=bFhOaG50K0E3OUZwZytJYUJnSGp6QT09

Meeting ID: 914 0401 0846

Password: 681668

**Speakers:**

- Raphaël Krikorian (Université de Cergy-Pontoise, France)
- Jairo Bochi (Pontificia Universidad Católica de Chile, Chile)
- Sylvain Crovisier (Université Paris-Sud 11, France)
- Mark Pollicott (University of Warwick, UK)

** Schedule** (all times are Lisbon times):

**13h Raphaël Krikorian**(Université de Cergy-Pontoise, France)

Title: On the divergence of Birkhoff Normal Forms

Abstract: A real analytic symplectic diffeomorphism of R^{2d} admitting a non resonant elliptic fixed point is always formally conjugated to a formal integrable system, its Birkhoff Normal Form (BNF). Siegel proved in 1954 that the involved formal conjugation does not in general define a converging series. I will give a proof of the fact that, in any dimension, the same phenomenon holds for the Birkhoff Normal Form itself (the formal integrable model). The key result is that the convergence of the BNF of a real analytic symplectic diffeomorphism of the plane has strong dynamical consequences on the diffeomorphism: the measure of the set of its invariant curves is abnormally large. In other words, an information on the formal dynamics has consequences on the "real" dynamics.

**14h Jairo Bochi**(Pontificia Universidad Católica de Chile, Chile)

Title: Finiteness of matrix equilibrium states

Abstract: I will discuss subadditive thermodynamical formalism. This theory originates from the study of dimensions of non-conformal fractals, but it is interesting by itself. Given a linear cocycle, we define a subadditive pressure, depending on parameters. By a subadditive variational principle, this pressure equals the supremum, over all invariant probability measures, of the metric entropy plus an appropriate linear combination of the Lyapunov exponents. Measures that attain the supremum are called equilibrium states. There are known sufficient conditions for the equilibrium state to be unique (and to have several nice ergodic properties), and these conditions are satisfied "generically". We are interested in the non-generic case. With Ian Morris, we proved that for every locally constant cocycle (of invertible matrices) over a full shift and every choice of parameters, the number of ergodic equilibrium states is finite, and we have a bound for their number. The proof is based on some simple ideas from algebraic geometry. I will conclude discussing possible extensions of these tools and results to more general classes of linear cocycles.

__15h-16h Break__**16h Sylvain Crovisier**(Université Paris-Sud 11, France)

Title: Spectral decomposition of surface diffeomorphisms

Abstract: In order to describe the dynamics of a diffeomorphism, one first decomposes the system into invariant elementary pieces, that may be studied separately. For hyperbolic dynamics, the decomposition is provided by Smale’s «spectral decomposition theorem».

This talk deals with the decomposition of general smooth diffeomorphisms on surface (several notions of pieces are natural). When the decomposition is infinite, one expects that the hyperbolicity degenerates along sequences of distinct pieces. I will discuss this problem in different cases, depending on the entropy of the system.

**17h Mark Pollicott**(University of Warwick, UK)

Title: Calculating Lyapunov exponents for random products of positive matrices.

Abstract: Given two (or more) square matrices, a natural quantity to study is the Lyapunov exponent associated to their random products. In most cases it isn't possible to give an explicit formula for the Lyapunov exponent and one has to resort to calculating its value numerically. We will discuss one particular method which works particularly well when the matrices have positive entries.

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**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.