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2 The BirkhoffKhinchin Ergodic Theorem Ergodicity
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absolutely continuous adjoint arbitrary assume Bernoulli automorphism billiards boundary Chap Choose circle S1 compact condition consider const constructed coordinates corresponding countable cyclic defined Definition denote diffeomorphism dynamical systems elements endomorphism equality equations equivalent ergodic theorem exists finite number follows formula function Gauss geodesic flow Gibbs group G Haar measure Hamiltonian Hence homeomorphism implies inequality integral intersection interval exchange transformation invariant measure invariant with respect Jm Jm Lebesgue measure Lebesgue space lemma is proved Let us prove Let us show manifold measure space metrically isomorphic mixing morphism obtain one-parameter group orthogonal particles partition phase space point x e point x0 Proof of Lemma properties r-algebra relation rotation number satisfying segment semi-interval sequence space L2(M spectral type spectrum statement subgroup subsets subspace Suppose theorem is proved theory torus Tor trajectory translation uniquely ergodic unitary operators vector field velocity