Invariant measure Billiard systems 2 Contraction maps 2 Countable iterated function system 2 Cycles 2 Embedding 2 Ergodicity 2 Fourier transform 2 Frequencies 2 Gibbs measures 2 Hamiltonian 2 Invariant set 2 Permutations 2 Random perturbations 2 Specifications 2 Time-series 2 Uniform ergodicity 2