Absolute continuity of hyperbolic invariant measures for endomorphisms

Author： Liu Pei-Dong   Shu Lin  

Publisher： IOP Publishing

ISSN： 0951-7715

Source： Nonlinearity, Vol.24, Iss.5, 2011-05, pp. : 1595-1611

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Abstract

We prove that, for a C2 non-invertible but non-degenerate map f on a compact Riemannian manifold without boundary, a hyperbolic invariant measure μ is absolutely continuous with respect to the Lebesgue measure on the manifold if, under a condition on the Jacobian of the map, the measure satisfies two entropy formulae for positive exponents [13] and negative exponents [7], respectively. This implies that the entropy production eμ(f) = 0 if and only if μ ≪ Leb.

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