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Ulcigrai, Corinna

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

Ann. Math. (2) 173, No. 3, 1743-1778 (2011).
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 \(\eta\).
Reviewer: Gabriel Soler López (Cartagena)

Cited in 2 Reviews
Cited in 17 Documents

MSC:

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

Keywords:

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

Citations:

Zbl 1130.37304
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\textit{C. Ulcigrai}, Ann. Math. (2) 173, No. 3, 1743--1778 (2011; Zbl 1251.37003)
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