SESIÓN DOBLE del seminario de sistemas dinámicos.

 Salón de seminarios del IMERL (101)


Maik Gröger (Jagiellonian University)

Título: Fractal dimensions of continuum real trees of circle maps and 
their graphs.

Resumen: The talk will begin by recalling some classical and more 
recent results on the fractal dimensions of the classical Weierstrass 
functions. We will move on with some details of the Brownian continuum 
tree, in particular, how it can be obtained by a change of metric from 
a contour function on the unit interval. In fact, this metric change 
can be applied to all contour functions (or continuous circle 
mappings), which means that we can associate to all these functions a 
real (rooted) tree. I will elaborate on a result of Picard from 2008, 
who proved that the dimension theory of these trees is connected to 
their associated contour function: the upper box dimension of real 
trees coincides with the variation index of their contour function.
In the course of the talk I will show that this connection can be 
deepened by providing more results which link fractal dimensions of 
the tree and the graph of its contour function.
This is joint work (in progress) with S. Troscheit.


Mateo Ghezal (Paris-Sacley)

Title : Measures of maximal entropy for smooth local diffeomorphisms 
of surface

Resume : In 1989, Newhouse proved in a famous work that systems, on 
compact manifolds, which are infinitely differentiable admits measures 
of maximal entropy. He then conjectured that this number is finite. 
For surface diffeomorphisms it remained open until Buzzi, Crovisier 
and Sarig proved it two years ago, using techniques from symbolic 
dynamics and Yomdin theory. In higher dimension and for endomorphisms, 
where a lot of different behavior can appear, the conjecture is wrong 
in the general case. Nevertheless, we do not lose hope and we will see 
that, with some reasonable hypothesis, the conjecture is still true 
for surface maps without critical points. We will also discuss, if the 
time allows us, in which setting we also believe that our techniques 
can be used, and what are the obstructions and constrains in the 
general case.


Próximo viernes 06/09 Sergi Burniol (FING - Udelar).