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Un exemple de transformation dilatante et C1 par morceaux de l'intervalle, sans probabilité absolument continue invariante

Published online by Cambridge University Press:  19 September 2008

P. Gora  and
B. Schmitt
P. Gora
Affiliation:
Warsaw University, Institute of Mathematics, PKIN IX p, 00-901 Warsaw, Poland
B. Schmitt
Affiliation:
Département de Math., UFR Sciences et Techniques, Universite de Dijon, 21000 Dijon, France
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Abstract

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We construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).

So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?

Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 1 , March 1989 , pp. 101 - 113
Copyright
Copyright © Cambridge University Press 1989

References

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