How to prune a horseshoe

Carvalho, André de; Hall, Toby

doi:10.1088/0951-7715/15/3/201

Cited by 27 publications

(12 citation statements)

References 69 publications

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“…The dynamics of the DK flows appears to be reminiscent of the analysis of the dynamics of Lorenz attractors, as discussed for example in the survey by Ghys [18]. Moreover, the variation of the horseshoes for a smooth family for generic DK-flows with positive entropy suggests a comparison with the degeneration in the dynamics of the Lorenz attractors as studied by de Carvalho and Hall [9,10,11,19]. The analogy between the dynamics of a family of generic DK-flows and a family of Lorenz attractors suggests that the topic is worth further investigation.…”

Section: Derived From Kuperberg Flowsmentioning

confidence: 97%

Perspectives on Kuperberg flows

Hurder¹,

Rechtman²

2016

Preprint

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The "Seifert Conjecture" asks, "Does every non-singular vector field on the 3-sphere S 3 have a periodic orbit?" In a celebrated work, Krystyna Kuperberg gave a construction of a smooth aperiodic vector field on a plug, which is then used to construct counter-examples to the Seifert Conjecture for smooth flows on the 3-sphere, and on compact 3-manifolds in general. The dynamics of the flows in these plugs have been extensively studied, with more precise results known in special "generic" cases of the construction. Moreover, the dynamical properties of smooth perturbations of Kuperberg's construction have been considered. In this work, we recall some of the results obtained to date for the Kuperberg flows and their perturbations. Then the main point of this work is to focus attention on how the known results for Kuperberg flows depend on the assumptions imposed on the flows, and to discuss some of the many interesting questions and problems that remain open about their dynamical and ergodic properties. Contents

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Section: Derived From Kuperberg Flowsmentioning

confidence: 97%

Perspectives on Kuperberg flows

Hurder¹,

Rechtman²

2016

Preprint

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“…Clearly, the pruning front is expected to play a similar role as the critical point in the kneading theory. A mathematical formulation has been developed in [50,51].…”

Section: Pruning Of the Horseshoementioning

confidence: 99%

Toward pruning theory of the Stokes geometry for the quantum Hénon map

Shudo

Ikeda

2016

Nonlinearity

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The Stokes geometry for the propagator of the quantum Hénon map is studied in the light of recent developments of the exact WKB analysis. As the simplest possible situation the Hénon map satisfying the so-called horseshoe condition is closely analyzed, together with listing up local bifurcation patterns of the Stokes geometry. This is exactly in the same spirit as pruning theory for the classical horseshoe system, and the present paper is placed as the first step to establish pruning theory of the Stokes geometry for chaotic systems. Our analysis reveals that the birth and death of the WKB solutions caused by the Stokes phenomenon do not occur in a local but entirely global manner, reflecting topological nature encoded in the Stokes geometry. We derive an explicit general formula to enumerate the number of WKB solutions in the asymptotic region and obtain its growth rate, which is shown to be less than the topological entropy of the corresponding classical dynamics. The relations to the diffraction catastrophe integrals and one-step multiply folding maps are also discussed.

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“…Then pruning family P(F ) is defined to be the closure, in the C 0 -topology, of the set of all the pruning homeomorphisms of F . It can be shown that P(F ) contains uncountably many different models of dynamics [8,9] and the PFC states that P(F ) contains enough models to describe all Hénon maps.…”

Section: Pruningmentioning

confidence: 99%

Proof of the Pruning Front Conjecture for certain Hénon parameters

Mendoza

2013

Nonlinearity

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The Pruning Front Conjecture is proved for an open set of Hénon parameters far from unimodal. More specifically, for an open subset of Hénon parameter space, consisting of two connected components one of which intersects the area-preserving locus, it is shown that the associated Hénon maps are prunings of the horseshoe. In particular, their dynamics is a subshift of the two-sided two-shift. A different way to formalize pruning for the Lozi family was introduced by Ishii in [13] which enabled him to prove the PFC for that family.

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How to prune a horseshoe

Cited by 27 publications

References 69 publications

Perspectives on Kuperberg flows

Perspectives on Kuperberg flows

Toward pruning theory of the Stokes geometry for the quantum Hénon map

Proof of the Pruning Front Conjecture for certain Hénon parameters

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