BEHAVIORS OF ENTROPY ON FINITELY GENERATED GROUPS

• 成果类型: Article
• 署名作者: Brieussel, Jeremie
• 署名单位: Kyoto University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP761
• 发表日期: 2013
• 页码: 4116-4161
• 关键词: random-walks; GROWTH; amenability; Automata

摘要

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any 1/2 <= alpha <= beta <= 1, there is a group Gamma with measure mu equidistributed on a finite generating set such that lim inf log H-Gamma,H-mu(n)/log n = alpha, lim sup H-Gamma,H-mu(n)/log n = beta The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk Y-n on such groups are also evaluated, providing an example of group with return probability satisfying lim inf log vertical bar log P(Y-n =(Gamma) 1)vertical bar/log n = 1/3, lim sup log vertical bar log P(Y-n =(Gamma) 1)vertical bar/log n = 1 and drift satisfying lim inf log E parallel to Y-n parallel to/log n = 1/2, lim sup log E parallel to Y-n parallel to/log n = 1.