# Closing session of the 1st International Online GSDUAB Seminar

- Date:
- 31.05.21
- Place:
- Online seminar
- Speaker:
- Yulij S. Ilyashenko, Jaume Llibre, Michal Misiurewicz
- Web Site:
- http://zoom.us/j/95097039724?pwd...

**Schedule**

- 12:00 (UTC+2) "Results and open problems on the algebraic limit cycles of the planar polynomial differential systems",
**Jaume Llibre**(Universitat Autònoma de Barcelona) - 15:00 (UTC+2) "Large bifurcation supports",
**Yulij S. Ilyashenko**(NRU - Higher School of Economic) - 16:10 (UTC+2) "Flexibility of entropies for piecewise expanding unimodal maps",
**Michal Misiurewicz**(Indiana University-Purdue University Indianapolis)

**Abstracts**

**Results and open problems on the algebraic limit cycles of the planar polynomial differential systems**[Jaume Llibre]

In this talk we summarize some results and open problems on the algebraic limit cycles of the planar polynomial differential systems. More precisely,

- we study the maximum number of algebraic limit cycles of the polynomial differential differential systems of degree n;
- we show how to use the algebraic limit cycles for proving that any finite configuration of limit cycles can be realized by some polynomial differential system;
- we provide the maximum number of algebraic limit cycles formed by circles that a polynomial differential system of degree $n$ can exhibit;
- we study the algebraic limit cycles of the polynomial differential systems of degree 2.

** Large bifurcation supports** [Yulij S. Ilyashenko]

Global bifurcation theory on the sphere is now in the study of its creation. The subject of the talk is: given a degenerate vector field, determine, what part of its phase portrait actually bifurcates? The talk is devoted to an answer to this question obtained in a joint work with Natalya Goncharuk. This answer is expected to be a powerful tool in the study of classification and structural stability of generic families of vector fields on the sphere with an arbitrary number of parameters.

** Flexibility of entropies for piecewise expanding unimodal maps** [Michal Misiurewicz]

We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most log 2, and the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 different slopes. For those maps, there is an additional restriction that the topological entropy is larger than (1/2) log 2. Moreover, we generalize and give a different interpretation of the Milnor-Thurston formula connecting the topological entropy and the kneading determinant for unimodal maps.

This is joint work with Lluís Alsedà and Rodrigo Pérez.