0000000000149631

AUTHOR

Dominique Malicet

showing 2 related works from this author

Entropy, Lyapunov exponents, and rigidity of group actions

2018

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teres\'opolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems…

Pure mathematicsPrimary 22F05 22E40. Secondary 37D25 37C85[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]Rigidity (psychology)Dynamical Systems (math.DS)Group Theory (math.GR)Mathematical proof01 natural sciencesMeasure (mathematics)[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]Group action0103 physical sciencesFOS: MathematicsErgodic theoryMSC : Primary: 22F05 22E40 ; Secondary: 37D25 37C850101 mathematicsAbelian groupMathematics - Dynamical SystemsEntropy (arrow of time)Mathematics[MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR]010102 general mathematicsLie group010307 mathematical physicsMathematics - Group Theory
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Ping-pong configurations and circular orders on free groups

2017

We discuss actions of free groups on the circle with "ping-pong" dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains or Markov partitions. Using this, we show that the free group $F_n$ admits an isolated circular order if and only if n is even, in stark contrast with the case for linear orders. This answers a question from (Mann, Rivas, 2016). Inspired by work of Alvarez, Barrientos, Filimonov, Kleptsyn, Malicet, Menino and Triestino, we also exhibit examples of "exotic" isolated points in the space of all circular orders on $F_2$. Analogous results are obtained for linear orders on the groups $F_n \times \mathbb{Z}$.

[ MATH.MATH-GR ] Mathematics [math]/Group Theory [math.GR][ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]MSC2010: Primary 20F60 57M60. Secondary 20E05 37C85 37E05 37E10 57M60.Extension (predicate logic)Group Theory (math.GR)Dynamical Systems (math.DS)Space (mathematics)20F60 57M60[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]CombinatoricsFree groupsOne-dimensional dynamicsFree groupPing pongFOS: MathematicsDiscrete Mathematics and CombinatoricsOrder (group theory)Geometry and TopologyMathematics - Dynamical SystemsMathematics - Group TheoryMathematicsOrders on groups
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