9th One Day Meeting on Dynamical

Systems

Day meeting

7 Jul 2020, via Zoom

The main topic of this event is the Lyapunov exponents in their various manifestations in Dynamical Systems along with their applications to other areas of mathematics.

**UPDATE:** Click here for the videos and slides of the meeting.

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.

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.

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.

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.

Time (GMT+1) | 7th July |
---|---|

13:00 | Raphaël Krikorian |

14:00 | Jairo Bochi |

15:00 | Break |

16:00 | Sylvain Crovisier |

17:00 | Mark Pollicott |

This event will be held virtually using Zoom. All registered participants will receive an email with the Zoom link to join the event.