Infinite measures, self-similar actions and continued fractions in ergodic theory

• Grant number: 11/12133-0
• Support Opportunities: Scholarships abroad - Research
• Start date: September 27, 2011
• End date: February 23, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Albert Meads Fisher
• Grantee: Albert Meads Fisher
• Host Investigator: Vadim Kaimanovitch
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Institution abroad: University of Ottawa (uOttawa), Canada
• Associated research grant: 06/03829-2 - Dynamic in low dimensions, AP.TEM

Abstract

The principal objective is to deepen our understanding of self-similar objects, making use of ideas from dynamical systems and ergodic theory, including the ergodic theory of infinite measures invariant for transformations and for group actions. A central theme is the study of connections between several apparently quite different areas: the theory of non-amenable groups, of Julia sets, of interval exchange transformations, and infinite measure ergodic theory. Links between these areas can be seen using the notions of "scenery flow", developed in [BedfordFisher96], [BedfordFisherUrbanski02], [Fisher04], of the "scaling functions" defined by Sullivan on his dual Cantor set [Sullivan87], [BedfordFisher97], of the self-similarity and nonstationary dynamics modelled in [ArnouxFisher05], and in the subject of self-similar groups, an area recently developed by Grigorchuk, Nekrashevych and Bartholdi among others [BartholdiGrigorchukNekrashevych2003], [Nekrashevych2005], [Nekrashevych2006]. (AU)