Topološki seminar Ljubljana - Maribor - Zagreb

Vodje seminarja: Petar Pavešić (Ljubljana), Iztok Banič (Maribor), Sonja Štimac (Zagreb)

Sliki sodelujočih: Slika1 Slika2

Bodoča predavanja

Minula predavanja

Lozi-like maps
Predavateljica: Sonja Štimac (Sveučilište u Zagrebu)

I will talk about Lozi-like family of maps, which is a generalization of the Lozi family of maps, and contains the Lozi family as a subfamily. The kneading theory developed for the Lozi maps holds in the same way for the Lozi-like maps: an itinerary of every point of the Lozi attractor is completely characterized by the set of kneading sequences.

While the situation may appear similar to what we see in one dimension for unimodal maps, there is a big difference. The analogue for the Lozi family is the family of the tent maps. There we have one parameter and one kneading sequence. For the Lozi family we have two parameters, but infinitely many kneading sequences. Thus, by using concrete formulas, we immensely restrict the possible sets of kneading sequences. It makes sense to conjecture that in a generic n-parameter subfamily of Lozi-like maps, n kneading sequences determine all other kneading sequences. In fact, in an example at the end of my talk, we will see that two kneading sequences of the Lozi map determine the parameter values, and thus all kneading sequences.

This is joint work with Michal Misiurewicz.

Unimodal category
Predavatelj: Dejan Govc (Univerza v Ljubljani)

The concept of unimodal category [1] was first introduced by Baryshnikov and Ghrist in 2007. It is an invariant of functions $f:Xto[0,infty)$, similar in spirit to the Lyusternik-Schnirelmann category. I will give an overview of the basic definitions, known results and a monotonicity conjecture from [1]. Then I will discuss this conjecture in more detail.

As is turns out, the conjecture holds for functions on $X=mathbb R$ as well as for certain well-behaved functions on $X=mathbb R^n$, but fails for more general functions.

[1] Y. Baryshnikov and R. Ghrist, Unimodal category and topological statistics. Proceedings of NOLTA, 2011.

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