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Evolution of measures in the phase space of nonlinear Hamiltonian systems. (English) Zbl 1178.37049
Theor. Math. Phys. 136, No. 3, 1325-1335 (2003); translation from Teor. Mat. Fiz. 136, No. 3, 496-506 (2003).
Summary: We establish the existence of weak limits of solutions (in the class $$L _{p}, p\geq 1$$) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. We find the limit probability distributions in the configuration space. We give conditions for a uniform distribution of Gibbs ensembles for geodesic flows on compact manifolds.

##### MSC:
 37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010) 82C05 Classical dynamic and nonequilibrium statistical mechanics (general) 37A30 Ergodic theorems, spectral theory, Markov operators 70H05 Hamilton’s equations
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