Most expanding maps have no absolutely continuous invariant measure

ArticleOriginal scientific text

Title

Authors ¹

Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152, U.S.A.

We consider the topological category of various subsets of the set of expanding maps from a manifold to itself, and show in particular that a generic

C^{1}

expanding map of the circle has no absolutely continuous invariant probability measure. This is in contrast with the situation for

C^{2}

or

C^{1 + ε}

expanding maps, for which it is known that there is always a unique absolutely continuous invariant probability measure.

C^{1}

expanding map, Ruelle-Perron-Frobenius operator

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