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Zbl 1251.37003
Ulcigrai, Corinna
Absence of mixing in area-preserving flows on surfaces. (English)
[J] Ann. Math. (2) 173, No. 3, 1743-1778 (2011). ISSN 0003-486X; ISSN 1939-0980/e

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 $\eta$.
[Gabriel Soler López (Cartagena)]

MSC 2000:: *37A25 Ergodicity, mixing, rates of mixing
37E35 Flows on surfaces
37C40 Smooth ergodic theory, invariant measures

Keywords: area preserving flow; interval exchange transformation; mixing; Hamiltonian flow

Citations: Zbl 1130.37304

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