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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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