Title: Multifractal analysis in Dynamical systems I
报告人: 廖灵敏(Lingmin Liao) LAMA UMR 8050 CNRS, Universit‘e Paris-Est Cr‘eteil
报告时间：2016年7月27日15:00-17:00
报告地点：维格堂319
摘要
Birkhoff averages of a (real-valued) function in a given dynamical system are widely studied.
From multifractal analysis point of view, one is interested in the sizes (Hausdorff dimension) of the level sets of the Birkhoff averages.
In this first talk, we begin with a classical result of Besicovitch (1934) and Eggleston (1949) on the Hausdorff dimension of the set of numbers with prescribed frequency of the digit 1 in its dyadic expansion. Then we present the corresponding result in continued fractions obtained by Fan, Liao and Ma (2010). By using Ruelle transfer operator theory, results on some special Birkhoff averages in continued fractions will also be obtained (Fan, Liao, Wang and Wu 2009). The results on the continued fractions can be generalized to interval maps (Fan, Jordan, Liao and Rams 2015). Finally, we will also discuss about the Birkhoff sums normalized by a fast increasing function. Some recent results including some works in progress will be mentioned.
Title: Multifractal analysis in Dynamical systems II
报告人: 廖灵敏(Lingmin Liao) LAMA UMR 8050 CNRS, Universit‘e Paris-Est Cr‘eteil
报告时间：2016年7月28日09:00-11：00
报告地点：维格堂319
摘要
Convergences of the multiple ergodic averages in dynamical systems recently attract much attention in ergodic theory (Fustenberg 1977, Conze--Lesigne 1984-1988, Bourgain 1990, Host--Kra 2001, Tao 2008). In this second talk, we are interested in the level sets of some multiple ergodic averages in dynamical systems. We first consider the full shift on the alphabet ${-1,1}$. We obtain a general formule for the Hausdorff dimension of such level sets by using Riesz products (Fan, Liao and Ma 2012). The same question is much more difficult for the full shift on the alphabet ${0,1}$. To solve the problem, some kind of non-linear Ruelle operator will be used (Kenyon, Peres and Solomyak 2012; Fan, Schmeling, Wu 2016). The recents result of Liao and Rams 2013, and Fan, Liao and Wu will also be discussed.