Dynamics Seminar “A midsummer dynamic's daydream”

Relatori

    Dong Han Kim
    Dongguk University - Seoul
    Maria Josè Pacifico
    Federal University of Rio de Janeiro
    Jacopo De Simoi
    Toronto University

9:15-10:15

Dong Han Kim

(Dongguk University - Seoul)

Title: "Intrinsic Diophantine approximation on circles and spheres”

Abstract: We study Lagrange spectra arising from intrinsic Diophantine approximation of copies of a circle and a sphere. More precisely, we consider three copies of a unit circle embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$ and three copies of a unit sphere embedded in $\mathbb{R}^3$ or $\mathbb{R}^4$. We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of $\mathbb{R}$ and $\mathbb{C}$. Combining this with prior work of Asmus L.~Schmidt on the spectra of $\mathbb{R}$ and $\mathbb{C}$, we can characterize, for each of the six spectra, the smallest accumulation point and the initial discrete part leading up to it completely. This is joint work with Byungchul Cha.

 

10:15 -10:30

Break

 

10.30-11:30

Maria Josè Pacifico

(Federal University of Rio de Janeiro)

Title: “Uniqueness of equilibrium states for Lorenz attractors”

Abstract: In this talk we consider the thermodynamic formalism for Lorenz attractors of flows in any dimension. Under a mild condition on the H ̈older continuous potential function φ, we prove that for an open and dense subset of C1 vector fields, every Lorenz attractor supports a unique equilibrium state.

In particular, we obtain the uniqueness for the measure of maximal entropy. This is joint work with Fan Yang and Jiagang Yang.

 

11:30 -11:45

Break

 

11:45 - 12:45

 Jacopo De Simoi 

(Toronto University)

Title: “Length spectrum of smooth convex billiards”

Abstract: Given a (convex) billiard table, we record the lengths of all periodic orbits in a set called the "Length spectrum"; we can then ask how much of the Geometry of the domain is encoded in the Length Spectrum.  This question is tightly related to the analogous question for the Spectrum of the Laplace operator, that is known as "Can one hear the shape of a drum?". It is known that a marking of the length spectrum (i.e. knowing "which" orbit corresponds to "which" length) allows to gather lots of dynamical information (e.g. Lyapunov exponents of periodic orbits and, in some cases, the whole geometry of the domain).  To which extent can such results be obtained without a marking? Does the length spectrum has any structure that can be used to recover a marking? In this talk I will show some evidence that this task can be quite complicated: we construct (a dense set of) smooth billiard domains with a very degenerate (uncountable) Length Spectrum.

 

WEB SITE:  http://www.crm.sns.it/course/6278/

Those who are unable to be physically present may attend the talk via Zoom by following the link: https://us02web.zoom.us/j/83965821067?pwd=YU9MRHVVMlRQMXZycjVzVGR3VmlXZz09