Examples of expandingC 1 maps having no σ-finite invariant measure equivalent to Lebesgue

In this paper we construct aC ¹ expanding circle map with the property that it has no σ-finite invariant measure equivalent to Lebesgue measure. We extend the construction to interval maps and maps on higher dimensional tori and the Riemann sphere. We also discuss recurrence of Lebesgue measure for the family of tent maps. Supported by the Deutsche Forschungsgemeinschaft (DFG). The research was carried out while HB was employed at the University of Erlangen-Nürnberg, Germany. Partially supported by NSF grant DMS # 9203489.