Dynamical Systems

Dynamical system and representation theory

In this talk, a practical method is described for computing the classical normal form of vector fields near the bifurcation point. Some necessary formulas are derived and applied to the anharmonic oscillator, the Bogdanov-Takens bifurcation, the 3D nilpotent problem, and elastic pipe conveying fluid,  to demonstrate the applicability of the theoretical results.
 Then, a review will be given of the developments in the last decade concerning the classification of unique normal forms in 3D nilpotent problems.

From time-average replicator to best-response dynamics, and back

When a game is played over time, the players can change their actions throughout the game. The choice of action can follow different learning mechanisms leading to different types of dynamics.    In this talk I shall look at the relation between the time-average of replicator dynamics (RD) and best-response dynamics (BRD), and its corresponding fictitious play (FP).  It is known that the time-average of RD converges to an invariant set under BRD but not whether given an invariant set BRD there always exists a corresponding RD orbit.

Parabolic Flows Renormalised by Partially Hyperbolic Maps

We will discuss 3-dimensional parabolic flows which are renormalised by circle extensions of Anosov diffeormorphisms (this includes nilflows on the Heisenberg nilmanifold). We use the spectral information of the transfer operators associated to these partially hyperbolic maps to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow. (Joint work with Lucia Simonelli.)

A formula for the local metric pressure

This talk starts with a generalization of the formula of Brin and Katok concerning the local metric entropy. The new formula provides a simple proof of the fact that invariant weak-Gibbs probability measures are equilibrium states. It also clarifies the role of the topological pressure on the standard definition of Gibbs property. This is a joint work with Sebastián Pérez.

Non-hyperciclicity for certain classes of linear dynamical systems

The investigation of the properties of bounded linear maps on certain vector spaces (Hilbert or Banach spaces, for example) is a very rich and active area. In particular, the existence of dense orbits (that in this context is known as hyperciclicity) attracts a lot of attention, as well as the extension of classical results to this setting, like hyperbolicity and shadowing, among many others.

Dynamics of piecewise $\lambda$-affine contractions

In this talk we discuss the dynamics of interval piecewise $\lambda$-affine contractions $f_\lambda$ where $\lambda\in(0,1)$. The map $f_\lambda$ is called asymptotically periodic if the $\omega$-limit set of every point is a periodic orbit and there are at most a finite number of periodic orbits. Under a generic assumption on $f_\lambda$, we show that the set of $\lambda\in(0,1)$ such that $f_\lambda$ is not asymptotically periodic has zero Hausdorff dimension.

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