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
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)