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On diffeomorphisms with polynomial growth of the derivative on surfaces

Krzysztof Frączek (2004)

Colloquium Mathematicae

We consider zero entropy C ∞ -diffeomorphisms on compact connected C ∞ -manifolds. We introduce the notion of polynomial growth of the derivative for such diffeomorphisms, and study it for diffeomorphisms which additionally preserve a smooth measure. We show that if a manifold M admits an ergodic diffeomorphism with polynomial growth of the derivative then there exists a smooth flow with no fixed point on M. Moreover, if dim M = 2, then necessarily M = ² and the diffeomorphism is C ∞ -conjugate to a skew...

On disjointness properties of some smooth flows

Krzysztof Frączek, Mariusz Lemańczyk (2005)

Fundamenta Mathematicae

Special flows over some locally rigid automorphisms and under L² ceiling functions satisfying a local L² 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 ∙ special flows built over ergodic interval exchange...

On symmetric logarithm and some old examples in smooth ergodic theory

K. Frączek, M. Lemańczyk (2003)

Fundamenta Mathematicae

We give a positive answer to the problem of existence of smooth weakly mixing but not mixing flows on some surfaces. More precisely, on each compact connected surface whose Euler characteristic is even and negative we construct smooth weakly mixing flows which are disjoint in the sense of Furstenberg from all mixing flows and from all Gaussian flows.

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