Self-similarity and the transition from finite to infinite measures in dynamical systems

• Grant number: 15/00037-7
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: February 16, 2015
• End date: February 15, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Albert Meads Fisher
• Grantee: Albert Meads Fisher
• Visiting researcher: Marina Talet
• Visiting researcher institution: Aix-Marseille Université (AMU), France
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 11/16265-8 - Low dimensional dynamics, AP.TEM

Abstract

The central theme of this project is to investigate asymptotically self-similar return time behavior for three quite different types of dynamical systems.(1) We study a parametrized family of maps of the interval with an indifferent fixed point; the main objective is to complete a paper about this together with A. Lopes and M. Talet. We see three distinct phases of behavior for the family,depending on the value of the parameter , and governed by a self-similar process: a Brownian motion for ± >2, a completely asymmetric ±-stable process for 1< ±<2, and a Mittag-Leffler process of index ± for 0< ±<1. More precisely, in all cases, we prove an almost-sure invariance principle in log density (asip (log)) for the number of returns to a subinterval. In the Gaussian phase the expected return time and the variance are both finite; in the stable phase the variance has become infinite, and in the Mittag-Leffler phase both the expected return time and the natural invariant measure are infinite. In this last phase the return-time sets are an integer fractal set of dimension ± in the sense of Bedford and Fisher, Proc. London Math. Soc.'92, and we use the asip (log)to prove for this phase an order-two ergodic theorem in the sense of Aaronson, Denker, and Fisher, Proc. AMS, '92.This paper completes a circle of papers begun by a study of the phases 1< ±<2 and ±>2 in Fisher and Lopes, Nonlinearity '01, where polynomial decay of correlation was shown, with distributional convergence to the Gaussian (i.e.~a Central Limit Theorem) being proved for ± >2. Here however we use completely different methods, developed in a series of three papers by Fisher and Talet, Annales de l'IHP, Prob-Stat '12,Electronic Journal Prob, '11, and Journal d'Analyse Mathématique '14. (2) In this part we encounter a very different class of examples with infinite measures and fractal return-time sets.The immediate objective here is to complete two articles, in the first of which we study Vershik's adic transformations,giving a complete classification of the invariant measures which are finite on some sub-Bratteli diagram. This extends and strengthens theorems of the authors, Fisher, Stochastics and Dynamics '09, and Ferenczi, Fisher, Talet, Journal d'Analyse Mathématique '09, as well as theorems in two papers of Bezuglyi, Kwiatkowski, Medynets, Solomyak '10, '11. In the second new paper we apply these results to give a classification of cutting-and-stacking transformations; we introduce an especially interesting class of examples, nested circle rotations, for which we give a necessary and sufficient condition for the finite ness of the measure, while proving that for the case of periodic combinatorics, one has fractal return times in the sense that one can prove an order-two ergodic theorem. This last result builds on Fisher ETDS '92, as well as Medynets and Solomyak '14. A further objective is to extend this last result beyond the periodic case.(3) Here the goal is to prove some related theorems for Brownian motion in a drifted Brownian medium, in dimension one. The model has been studied since 1982, and exhibits two levels of randomness. At fixed environment, the quenched case, one has a Markov process. But after averaging over the randomness of the environment, the annealed case, the process is in general no longer Markov.Annealed limit theorems were proved in Brox'82 in the recurrent drift less case and by Kawazu and Tanaka '96, '97, '98, Hu, Shi and Yor '99, in the transient nonzero drift case. In the annealed and quenched settings, large deviations results were obtained in Talet Ann. Prob. '01 and Ann Prob.'07, and moderate deviations in Hu and Shi '04.In view of this work, we have recently proved some intermediate results leading us to believe that one can build on the methods in Fisher-Talet '12 to obtain analogous results. (AU)