EVENT DETAILS AND ABSTRACT

Title: Multifractal analysis for multimodal maps
Speaker: Professor Mike Todd
Speaker Info: University of Porto
Brief Description:
Special Note:
Abstract:

Let $f$ be a multimodal map of the interval $I$. Given an equilibrium state $\mu_\phi$ for a H\"older potential $\phi:I \rightarrow R$, the local dimension $d_{\mu_\phi}(x)$ measures how concentrated $\mu_\phi$ is at this point. The dimension spectrum encodes the Hausdorff dimension of level sets of $d_{\mu_phi}$. This spectrum can be understood via induced maps $(X, F)$, where $F = f^\tau$ for some inducing time $\tau$. A major challenge for maps with critical points is to find inducing schemes which `see' a sufficiently large subset of the space. In this talk I will explain that this problem can be overcome, and hence that, as in the uniformly expanding case, the dimension spectrum is encoded by a function related to the pressure of some potentials involving $\phi$. Many of these results apply not only to Collet-Eckmann maps, but also to maps with much weaker growth conditions.