Blender-producing mechanisms and a dichotomy for local dynamics near heterodimensional cycles

Abstract Blenders are special hyperbolic sets used to produce various robust dynamical phenomena that appear fragile at first glance. We prove that for <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>C</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:math> $C^r$ upper C Superscript r diffeomorphisms ( <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>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> </mml:math> $r=2,\ldots ,\infty ,\omega $ r equals 2 comma ellipsis comma infinity comma omega ), blenders naturally exist (without perturbation) near non-degenerate heterodimensional cycles of coindex 1, and the existence is determined by arithmetic properties of moduli of topological conjugacy for diffeomorphisms with heterodimensional cycles. In particular, we obtain a dichotomy for the dynamics in any small neighborhood U of a non-degenerate heterodimensional cycle: either there exist infinitely many blenders accumulating on the cycle, forming robust heterodimensional dynamics in most cases, or there are no orbits, other than those constituting the cycle, lying entirely in U .