# Linear growth of the derivative for measure-preserving diffeomorphisms

Colloquium Mathematicae (2000)

- Volume: 84/85, Issue: 1, page 147-157
- ISSN: 0010-1354

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topFrączek, Krzysztof. "Linear growth of the derivative for measure-preserving diffeomorphisms." Colloquium Mathematicae 84/85.1 (2000): 147-157. <http://eudml.org/doc/210793>.

@article{Frączek2000,

abstract = {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.},

author = {Frączek, Krzysztof},

journal = {Colloquium Mathematicae},

keywords = {ergodic -diffeomorphism; measure-preserving diffeomorphisms; irrational rotation},

language = {eng},

number = {1},

pages = {147-157},

title = {Linear growth of the derivative for measure-preserving diffeomorphisms},

url = {http://eudml.org/doc/210793},

volume = {84/85},

year = {2000},

}

TY - JOUR

AU - Frączek, Krzysztof

TI - Linear growth of the derivative for measure-preserving diffeomorphisms

JO - Colloquium Mathematicae

PY - 2000

VL - 84/85

IS - 1

SP - 147

EP - 157

AB - 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.

LA - eng

KW - ergodic -diffeomorphism; measure-preserving diffeomorphisms; irrational rotation

UR - http://eudml.org/doc/210793

ER -

## References

top- [1] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, Berlin, 1982.
- [2] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES 49 (1979), 5-234.
- [3] A. Iwanik, M. Lemańczyk and D. Rudolph, Absolutely continuous cocycles over irrational rotations, Israel J. Math. 83 (1993), 73-95. Zbl0786.28011
- [4] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974.

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