Accessibility and central integrability in the absence of periodic points

Abstract

<jats:p> We consider a partially hyperbolic diffeomorphism <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>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false" accent="false">→</mml:mo> <mml:mi>M</mml:mi> </mml:math> <jats:tex-math>$f: M \to M$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0143385726103563_inline1.png"> <jats:alt-text content-type="machine-generated">f colon upper M right arrow upper M</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> without periodic points on a closed manifold <jats:italic>M</jats:italic> . We prove that <jats:italic>f</jats:italic> is accessible when <jats:italic>M</jats:italic> is a 3-manifold with non-virtually-solvable fundamental group <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:msub> <mml:mi>π</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> <jats:tex-math>$\pi _1(M)$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0143385726103563_inline2.png"> <jats:alt-text content-type="machine-generated">pi 1 left parenthesis upper M right parenthesis</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> . In the case where <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>dim</mml:mi> <mml:mo>⁡</mml:mo> <mml:msup> <mml:mi>E</mml:mi> <mml:mi>c</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> <jats:tex-math>$\dim E^c = 1$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0143385726103563_inline3.png"> <jats:alt-text content-type="machine-generated">dimension upper E Superscript c Baseline equals 1</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> , we demonstrate that the center bundle <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:mi>E</mml:mi> <mml:mi>c</mml:mi> </mml:msup> </mml:math> <jats:tex-math>$E^c$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" content-type="simple" xlink:href="S0143385726103563_inline4.png"> <jats:alt-text content-type="machine-generated">upper E Superscript c</jats:alt-text> </jats:inline-graphic> </jats:alternatives> </jats:inline-formula> is uniquely integrable if <jats:italic>f</jats:italic> lacks accessibility. Furthermore, we provide a complete characterization of accessibility classes for such systems with one-dimensional center bundles. </jats:p>