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Absence of mixing in area-preserving flows on surfaces. (English) Zbl 1251.37003
This paper solves an open question by {\it A. Katok} and {\it 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 $\eta$.

37A25Ergodicity, mixing, rates of mixing
37E35Flows on surfaces
37C40Smooth ergodic theory, invariant measures
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