The transition from finite to infinite measures in dynamical systems

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Grant number: | 15/00037-7 |

Support type: | Research Grants - Visiting Researcher Grant - International |

Duration: | February 16, 2015 - 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 |

Home 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-similarreturn time behavior for three quite different types of dynamical systems.(1) We study a parametrized family of maps of the interval with anindifferent fixed point; the main objective is to complete a paperabout thistogether with A. Lopes and M. Talet. We see three distinct phases of behaviorfor the family,depending on the value of the parameter , and governed by a self-similar process: a Brownian motion for ± >2, a completelyasymmetric ±-stable process for1< ±<2, and a Mittag-Leffler process of index ± for 0< ±<1. More precisely, in all cases, we prove an almost-sure invarianceprinciple in log density (asip (log)) for the number of returns to a subinterval. In theGaussian phase the expected return time and the variance are bothfinite; 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 areinfinite. In this last phase the return-time sets are an integer fractal set ofdimension ± 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 phases1< ±<2 and ±>2 in Fisher and Lopes, Nonlinearity '01, where polynomial decay of correlationwas shown, with distributional convergence to the Gaussian (i.e.~aCentral 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 arefinite on some sub-Bratteli diagram. This extends and strengthens theorems of the authors, Fisher, Stochastics andDynamics '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 applythese results to give a classification of cutting-and-stackingtransformations; we introduce an especially interesting class ofexamples, nested circle rotations, for which we give a necessary andsufficient condition for the finiteness of the measure, while proving that for the case of periodic combinatorics, one has fractal returntimes in the sense that one can prove an order-two ergodictheorem. This last result builds on Fisher ETDS '92, as well as Medynets and Solomyak '14. A further objective is to extend thislast 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 ofrandomness. At fixed environment, the quenched case, one has a Markovprocess. 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 driftless case and by Kawazu and Tanaka '96, '97, '98, Hu,Shi and Yor '99, in the transient nonzero drift case. In theannealed and quenched settings, largedeviations 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 thatone can build on the methods in Fisher-Talet '12 to obtain analogous results. (AU)