Asymptotic Laws For Symbolic Dynamical Systems

. These are notes of a mini-course given in the Summer School on Symbolic Dynamics, Temuco (Chile), January 1997. The aim is to present some results concerning asymptotic limit laws for the occurrence of asymptotically rare events associated with some symbolic dynamical systems. The results will be mainly presented for shifts of finite type, but there will be left questions of whether these results remain valid for more general or possibly intrinsically different subshifts. Contents Introduction 1 1. Preliminaries 2 1.1. Bernoulli trials 2 1.2. Occurrence and Waiting Times 4 2. General Setup and Motivation 8 2.1. Return Maps and Expected Return Times 8 2.2. Asymptotically Rare Events 10 2.3. Known Results and Motivation 10 3. Shifts of Finite Type and Equilibrium States 17 3.1. Holder Potentials 17 3.2. Entropy and Pressure 18 3.3. Ruelle-Perron-Frobenius Operator 19 3.4. The Central Limit Theorem 20 3.5. Pianigiani-Yorke Measure 21 4. Point Processes and Convergence in Law 25 4.1. Co..