Topological Invariance of a Strong Summability Condition in One-Dimensional Dynamics

Li, Huaibin; Shen, Weixiao

doi:10.1093/imrn/rns105

Cited by 6 publications

(6 citation statements)

References 17 publications

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“…In the following theorem we summarize and complement results in [8,9], when restricted to unicritical maps. We state it in a stronger form than what is needed for this section.…”

Section: A Priori Bounds and Backward Contractionmentioning

confidence: 73%

“…This allows us to use the characterization for backward contracting maps of the summability condition given in [9]. We recall this result in Sect.…”

Section: Strategy and Organizationmentioning

confidence: 97%

“…To prove Theorem B, we show that in either case the map is backward contracting (part 1 of Theorem 1). This allows us to use the characterization for backward contracting maps of the summability condition given in [LS11]. We recall this result in §3, as part 2 of Theorem 1.…”

Section: Introductionmentioning

confidence: 94%

“…Backward contraction. In this section we recall the definition of "backward contraction" from [RL07], and compare it with a variant from [LS11].…”

Section: Preliminariesmentioning

confidence: 99%

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On Poincaré Series of Unicritical Polynomials at the Critical Point

Rivera-Letelier

Shen²

2013

Commun. Math. Stat.

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“…In the following theorem we summarize and complement results in [8,9], when restricted to unicritical maps. We state it in a stronger form than what is needed for this section.…”

Section: A Priori Bounds and Backward Contractionmentioning

confidence: 73%

“…This allows us to use the characterization for backward contracting maps of the summability condition given in [9]. We recall this result in Sect.…”

Section: Strategy and Organizationmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 94%

“…Backward contraction. In this section we recall the definition of "backward contraction" from [RL07], and compare it with a variant from [LS11].…”

Section: Preliminariesmentioning

confidence: 99%

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On Poincaré Series of Unicritical Polynomials at the Critical Point

Rivera-Letelier

Shen²

2013

Commun. Math. Stat.

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“…But in [LW06] the authors proved that a strengthened version of the Collet-Eckmann condition for multimodal maps is topologically invariant, see [LW06,Main Theorem], and also [Wan01], for a precise statement. See also [LS13] for a recent related result.…”

Section: Introductionmentioning

confidence: 92%

Topological invariance of the Collet–Eckmann condition for one-dimensional maps

2017

Nonlinearity

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This paper is devoted to study the topological invariance of several non-uniform hyperbolicity conditions of one-dimensional maps. In contrast with the case of maps with only one critical point, it is known that for maps with several critical points the Collet-Eckmann condition is not in itself invariance under topological conjugacy. We show that the Collet-Eckmann condition together with any of several slow recurrence conditions is invariant under topological conjugacy. This extends and gives a new proof of a result by Luzzatto and Wang that also applies to the complex setting.|Df n (f (c))| ≥ Cλ n .This type of condition was first introduced by P. Collet and J.-P. Eckmann in [CE83] providing a large class of S-unimodal maps of the interval having a finite absolutely continuous invariant measure. Further investigations by T. Nowicki, S. van Strien, and D. Sands proved the equivalence of those and several similar properties in the real S-unimodal setting, see [Now85,NS98] and see also [Prz98] for rational maps.Recall that two multimodal maps f : I → I and g : J → J are topologically conjugate, if there exists a homeomorphism h : I → J such that for each x ∈ I we have h(f (x)) = g(h(x)). Moreover, if both h and h −1

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On Summability Conditions for Interval Maps

2013

Bull. Aust. Math. Soc.

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Topological Invariance of a Strong Summability Condition in One-Dimensional Dynamics

Cited by 6 publications

References 17 publications

On Poincaré Series of Unicritical Polynomials at the Critical Point

On Poincaré Series of Unicritical Polynomials at the Critical Point

Topological invariance of the Collet–Eckmann condition for one-dimensional maps

On Summability Conditions for Interval Maps

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