Ergodic Theory and Dynamical Systems
- Ergodic Theory and Dynamical Systems / Volume 9 / Issue 01 / March 1989, pp 101-113
- Copyright © Cambridge University Press 1989
- DOI: http://dx.doi.org/10.1017/S0143385700004831 (About DOI), Published online: 19 September 2008
Research Article
Un exemple de transformation dilatante et C1 par morceaux de l'intervalle, sans probabilité absolument continue invariante
P. Goraa1 and B. Schmitta2
a1 Warsaw University, Institute of Mathematics, PKIN IX p, 00-901 Warsaw, Poland
a2 Département de Math., UFR Sciences et Techniques, Universite de Dijon, 21000 Dijon, France
Abstract
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..
(Received June 15 1987)
(Revised October 22 1987)
