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

Grant number: 11/12133-0
Support type:Scholarships abroad - Research
Effective date (Start): September 27, 2011
Effective date (End): February 23, 2012
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Albert Meads Fisher
Grantee:Albert Meads Fisher
Host: Vadim Kaimanovitch
Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
Local de pesquisa : 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)