Entropy continuity for interval maps with holes

Bandtlow, Oscar F.; Rugh, Hans Henrik

doi:10.1017/etds.2016.115

Cited by 3 publications

(2 citation statements)

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“…To deduce the Hölder continuity of H (with certain Hölder exponent), we may apply the elegant theory of Keller and Liverani [9] on stability of spectrum of one-parameter family of bounded linear operators to Φ a (acting on suitable Banach space equipped with a second norm); see [9, Corollary 1] for details. For example, Bandtlow and Rugh [3] adopted this approach. However, since our focus here is on the regularity of H rather than more detailed information on the spectrum properties of Φ a , and since we aim at identifying the best Hölder exponents of H, we choose to analyze H in a direct way instead of applying [9, Corollary 1] in our argument.…”

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confidence: 99%

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On fair entropy of the tent family

Gao¹,

Gao²

2021

DCDS

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The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues [13] recently, and discussed in detail for piecewise monotone interval maps. In particular, they showed that the fair entropy h(a) of the tent map fa, as a function of the parameter a = exp(htop(fa)), is continuous and strictly increasing on [ √ 2, 2]. In this short note, we extend the last result and characterize regularity of the function h precisely. We prove that h is 1 2 -Hölder continuous on [ √ 2, 2] and identify its best Hölder exponent on each subinterval of [ √ 2, 2]. On the other hand, parallel to a recent result on topological entropy of the quadratic family due to Dobbs and Mihalache [7], we give a formula of pointwise Hölder exponents of h at parameters chosen in an explicitly constructed set of full measure. This formula particularly implies that the derivative of h vanishes almost everywhere.

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confidence: 99%

“…For the complex quadratic family, the continuity of core entropy (defined for postcritically finite polynomials acting on its Hubbard tree) was proved by Tiozzo [15]. Pointwise Hölder continuity of topological entropy was also studied for family of interval maps with holes by Bandtlow and Rugh [3].…”

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confidence: 99%

On fair entropy of the tent family

Gao¹,

Gao²

2021

DCDS

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Introduction

Pollicott¹,

Urbański²

2017

Lecture Notes in Mathematics

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Diabolical Entropy

Dobbs

Mihalache

2019

Commun. Math. Phys.

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Milnor and Thurston's famous paper proved monotonicity of the topological entropy for the real quadratic family. Guckenheimer showed that it is Hölder continuous. We obtain a precise formula for the Hölder exponent at almost every quadratic parameter.Furthermore, the entropy of most parameters is proven to be in a set of Hausdorff dimension smaller than one, while most values of the entropy arise from a set of parameters of dimension smaller than one. * N.D. was supported by the ERC Bridges grant † N.M. was supported by the ERC AG COMPAS grant, CNRS semester and ANR LAMBDA 1 Douady, Hubbard and Sullivan had proven that the number of periodic orbits of some fixed period is monotonically decreasing, which implies the monotonicity of entropy. This result was unpublished, a later version was published by Douady [15].

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Entropy continuity for interval maps with holes

Cited by 3 publications

References 32 publications

On fair entropy of the tent family

On fair entropy of the tent family

Introduction

Diabolical Entropy

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