A new criterion of physical measures for partially hyperbolic diffeomorphisms

Abstract

We show that, for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> partially hyperbolic diffeomorphism, there is a full volume subset such that any Cesàro limit of any point in this subset satisfies the Pesin formula for partial entropy. This result has several important applications. First, we show that, for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1 plus"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^{1+}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> partially hyperbolic diffeomorphism with one dimensional center, there is a full volume subset such that either every point in this set belongs to the basin of a physical measure with nonvanishing center exponent or the center exponent of any limit of the sequence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over n EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts delta Subscript f Sub Superscript i Subscript left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>n</mml:mi> </mml:mfrac> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:munderover> <mml:msub> <mml:mi>δ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\frac 1n\sum _{i=0}^{n-1}\delta _{f^i(x)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is vanishing. We also prove that, for any diffeomorphism with mostly contracting center, it admits a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-neighborhood such that every diffeomorphism in a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> residual subset of this open set admits finitely many physical measures whose basins have full volume.