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On diffeomorphisms with polynomial growth of the derivative on surfaces

100%
Frączek K.
Colloquium Mathematicum
|
2004
|
tom 99
|
nr 1
75-90
EN
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 = 𝕋² and the diffeomorphism is $C^{∞}$-conjugate to a skew product on the 2-torus.
2
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Linear growth of the derivative for measure-preserving diffeomorphisms

100%
Frączek K.
Colloquium Mathematicum
|
2000
|
tom 84/85
|
nr 1
147-157
EN
We consider measure-preserving diffeomorphisms of the torus with zero entropy. We prove that every ergodic $C^{1}$-diffeomorphism with linear growth of the derivative is algebraically conjugate to a skew product of an irrational rotation on the circle and a circle $C^{1}$-cocycle. We also show that for no positive β ≠ 1 does there exist an ergodic $C^{2}$-diffeomorphism whose derivative has polynomial growth with degree β.
3
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On disjointness properties of some smooth flows

63%
Frączek K., Lemańczyk M.
Fundamenta Mathematicae
|
2005
|
tom 185
|
nr 2
117-142
EN
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 transformations and under functions of bounded variation are disjoint from mixing flows; ∙ ergodic components of flows coming from billiards on rational polygons are disjoint from mixing flows; ∙ 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.
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