One-day Meeting on Fractal Geometry in Budapest

Schedule:

• 10:00-10:45 Francois Ledrappier
  Title: Variational Principle for Anosov groups of matrices
  

  Abstract

  We consider a discrete group of $(d \times d)$-matrices. There is a geometric growth rate (the “Falconer dimension”) that generalizes the Poincaré exponent of discrete groups of $(2\times 2)$-matrices. Under some hyperbolicity assumptions, we relate this growth rate to the supremum of the “Lyapunov dimensions” associated to random walks on the group. This is joint work with P. Lessa (Montevideo) and a generalization of a recent paper by Jiao, Li, Pan, and Xu.

• 11:00-11:45 Sascha Troscheit
  Title: Continuum trees of real functions and their graphs
  

  Abstract

  The Brownian continuum tree (CRT) is an important random metric space that was extensively investigated in the 1990s. It can be constructed by a change of metric from a Brownian excursion function on $[0,1]$. This change of metric can be applied to all continuous circle mappings to give a continuum tree associated with the function.
  In 2008, Picard proved that analytic properties of the function are connected to the dimension theory of its tree: the upper box dimension of the continuum tree coincides with the variation index of the contour function. We will provide a short and direct proof of Picard’s theorem through the study of packings. The methods used will inspire different notions of variations and variation indices, and we will link the dimension theory of the tree with the dimension theory of the graph of its contour function.
  (Joint with Maik Gröger)

• 12:15-13:00 Michal Rams
  Title: Lyapunov spectrum of matrix cocycles
  

  Abstract

  I will give an introduction into calculation of the Lyapunov spectrum of $SL(2,R)$ matrix cocycles, presenting results we got with Lorenzo Diaz and Katrin Gelfert.

Organisers: Balázs Bárány, Károly Simon, István Kolossváry