The transition from finite to infinite measures in dynamical systems

• Grant number: 09/17358-0
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Duration: February 01, 2010 - December 30, 2010
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Albert Meads Fisher
• Grantee: Albert Meads Fisher
• Visiting researcher: Marina Talet
• Visiting researcher institution: Université de Provence, France
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 06/03829-2 - Dynamic in low dimensions, AP.TEM

Abstract

In this project we study infinite measures for dynamical systems where the occurrence of the infinite measure has a geometrical explanation in terms of a fractal structure of return-times. We are especially interested in understanding the transition from finite to infinite measure. There are two aspects to the project: the development of the necessary ergodic theory and probability tools, and the application of these to the study of interesting examples. The first objective of the planned visit is to finish writing and then submitting for publication the article “Log average ergodic theorems and asymptotic self-similarity for renewal flows” (currently with 45 pages). This work builds upon two recently submitted papers “Dynamical attraction to stable processes” (42p) and “Self-similar dynamics of renewal processes” (27 p). Our second objective is to complete a paper with Artur Lopes on a one parameter family of maps of the interval with an indifferent fixed point which exhibits a transition finite to infinite measure as the parameter of the system is changed. This work is based on the previously mentioned three papers. The next part of the project involves diffusions in random media, an area of expertise of Prof. Talet. Our starting point is the observation that phase changes which occur in the indifferent fixed point example are reminiscent of some behavior of diffusions and also random walks in random media. We hope to apply our ergodic theory and dynamics methods and insights in the study of this very different set of examples. The final part of the project involves some again very different dynamical systems: adic transformations, interval exchange transformations and monocycle flows where interesting infinite (hence non-classical) invariant measures related to fractal geometry can be found. We build on two papers published in 2009, “Nonstationary mixing and the unique ergodicity of adic transformations”, A. M. Fisher, (56 p, Stochastics and Dynamics) and “Some (non-)uniquely ergodic adic transformations” S. Ferenczi, A.M. Fisher and M. Talet. (27 p, Journal d'Analyse Mathématique). (AU)