On disjointness properties of some smooth flows

Krzysztof Fr/aczek¹ ; Mariusz Lemańczyk¹

¹ Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopina 12/18 87-100 Toruń, Poland

Fundamenta Mathematicae, Tome 185 (2005) no. 2, pp. 117-142.

Résumé

Special flows over some locally rigid automorphisms and under $L^2$ ceiling functions satisfying a local $L^2$ Denjoy–Koksma type inequality are considered. Such flows are proved to be disjoint (in the sense of Furstenberg) from mixing flows and (under some stronger assumption) from weakly mixing flows for which the weak closure of the set of all instances consists of indecomposable Markov operators. As applications we prove that$\bullet$ special flows built over ergodic interval exchange transformations and under functions of bounded variation are disjoint from mixing flows; $\bullet$ ergodic components of flows coming from billiards on rational polygons are disjoint from mixing flows; $\bullet$ smooth ergodic flows of compact orientable smooth surfaces having only non-degenerate saddles as isolated critical points (and having a “good” transversal) are disjoint from mixing and from Gaussian flows.

Détail

DOI : 10.4064/fm185-2-2

Keywords: special flows locally rigid automorphisms under ceiling functions satisfying local denjoy koksma type inequality considered flows proved disjoint sense furstenberg mixing flows under stronger assumption weakly mixing flows which weak closure set instances consists indecomposable markov operators applications prove bullet special flows built ergodic interval exchange transformations under functions bounded variation disjoint mixing flows bullet ergodic components flows coming billiards rational polygons disjoint mixing flows bullet smooth ergodic flows compact orientable smooth surfaces having only non degenerate saddles isolated critical points having transversal disjoint mixing gaussian flows