Absence of mixing in area-preserving flows on surfaces.(English)Zbl 1251.37003

This paper solves an open question by A. Katok and J.-P. Thouvenot; see, for example, Section 6.3.2 of [“Spectral properties and combinatorial constructions in ergodic theory”, in: Handbook of dynamical systems. Volume 1B. Amsterdam: Elsevier. 649–743 (2006; Zbl 1130.37304)]. The statement of the main result is the following: Let $$S$$ be a closed surface of genus $$g\geq 2$$ and let $$\Phi:\mathbb{R}\times S\rightarrow S$$ be a flow given by a multi-valued Hamiltonian associated to a smooth closed differential $$1$$-form $$\eta$$. If $$\Phi$$ has only simple saddles and no saddle loops homologous to zero then $$\Phi$$ is not mixing for a typical such form $$\eta$$.

MSC:

 37A25 Ergodicity, mixing, rates of mixing 37E35 Flows on surfaces 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems

Zbl 1130.37304
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References:

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