Semi-conjugacy rigidity for endomorphisms derived from Anosov on the 2-torus

Abstract

<jats:p> Let <jats:italic>f</jats:italic> be a non-invertible partially hyperbolic endomorphism on <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> <jats:tex-math>$\mathbb {T}^2$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0143385726103447_inline1.png"> <jats:alt-text content-type="machine-generated">double struck upper T squared</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> that is derived from a non-expanding Anosov endomorphism. Differing from the case of diffeomorphisms derived from Anosov automorphisms, there is no <jats:italic>a priori</jats:italic> semi-conjugacy between <jats:italic>f</jats:italic> and its linearization on <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> <jats:tex-math>$\mathbb {T}^2$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0143385726103447_inline2.png"> <jats:alt-text content-type="machine-generated">double struck upper T squared</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> . We show that <jats:italic>f</jats:italic> is semi-conjugate to its linearization if and only if <jats:italic>f</jats:italic> admits a partially hyperbolic splitting with two <jats:inline-formula> <jats:alternatives> <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>D</mml:mi> <mml:mi>f</mml:mi> </mml:math> <jats:tex-math>$Df$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0143385726103447_inline3.png"> <jats:alt-text content-type="machine-generated">upper D f</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> -invariant subbundles. Moreover, if we assume that <jats:italic>f</jats:italic> has an unstable subbundle, then the semi-conjugacy is exactly a topological conjugacy, and the center Lyapunov exponents of the periodic points of <jats:italic>f</jats:italic> coincide with its linearization. In particular, <jats:italic>f</jats:italic> is an Anosov endomorphism and the conjugacy is smooth along the stable foliation. For the case where <jats:italic>f</jats:italic> has a stable subbundle, there is still some rigidity in its stable Lyapunov exponents. However, we also give examples that admit a partially hyperbolic splitting with center subbundle, but the semi-conjugacy is indeed non-injective. Finally, we present some applications under the volume-preserving assumption. </jats:p>