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Mildly dissipative diffeomorphisms of the disk with zero entropy

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arxiv 2005.14278 v3 pith:NVDXJRSR submitted 2020-05-28 math.DS

Sylvain Crovisier , Enrique Pujals , Charles Tresser This is my paper · ORCID
classification math.DS
keywords diffeomorphismsentropydynamicsmapsparticularprovesystemszero
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We discuss the dynamics of smooth diffeomorphisms of the disc with vanishing topological entropy which satisfy the mild dissipation property introduced in [CP]. In particular it contains the H\'enon maps with Jacobian up to 1/4. We prove that these systems are either (generalized) Morse Smale or infinitely renormalizable. In particular we prove for this class of diffeomorphisms a conjecture of Tresser: any diffeomorphism in the interface between the sets of systems with zero and positive entropy admits doubling cascades. This generalizes for these surface dynamics a well known consequence of Sharkovskii's theorem for interval maps.

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  1. A Priori Bounds for H\'enon-like Renormalization

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    Regularly renormalizable Hénon-like maps with bounded combinatorics have uniformly controlled stretching of horizontal curves, so their one-dimensional profiles are precompact.

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