Hyperbolic absolutely continuous invariant measures for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msup>

Abstract

<jats:p> For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , we show, using the Ledrappier–Young entropy characterization of SRB measures for non-invertible maps, that if a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:math> map <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> of the interval or the circle has its Lyapunov exponent greater than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>r</mml:mi> </mml:mfrac> <mml:mo form="prefix">log</mml:mo> <mml:msub> <mml:mrow> <mml:mo>∥</mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>′</mml:mi> </mml:msup> <mml:mo>∥</mml:mo> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msub> </mml:mrow> </mml:math> on a set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> of positive Lebesgue measure, then it admits hyperbolic ergodic invariant measures that are absolutely continuous with respect to the Lebesgue measure. We also show that the basins of these measures cover <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> Lebesgue-almost everywhere. </jats:p>