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<p>A diffeomorphism <i>f</i> has a heterodimensional cycle if it displays two (transitive) hyperbolic sets <i>K</i> and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K'\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>K</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> with different indices such that the unstable set of <i>K</i> intersects the stable one of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(K'\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>K</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> and vice versa. We prove that it is possible to find robust heterodimensional cycles for families of polynomial automorphisms of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {C}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. This provides a new example of an open set of non-hyperbolic systems in complex dynamics. The proof is based on Bonatti–Díaz blenders.</p>

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Robust complex heterodimensional cycles

  • 期刊论文

  • 发表日期：

    2026年3月15日

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  • Sébastien Biebler

摘要

A diffeomorphism f has a heterodimensional cycle if it displays two (transitive) hyperbolic sets K and \(K'\) K ′ with different indices such that the unstable set of K intersects the stable one of \(K'\) K ′ and vice versa. We prove that it is possible to find robust heterodimensional cycles for families of polynomial automorphisms of \(\mathbb {C}^3\) C 3 . This provides a new example of an open set of non-hyperbolic systems in complex dynamics. The proof is based on Bonatti–Díaz blenders.