On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

Johannes Kautzsch; Marc Kesseböhmer; Tony Samuel

doi:10.1007/s00023-015-0451-8

On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

Johannes Kautzsch, Marc Kesseböhmer, Tony Samuel

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We consider a family {T_r : [0, 1] → [0, 1] }_{r ∈ [0, 1]} of Markov interval maps interpolating between the tent map T₀ and the Farey map T₁. Letting P_r denote the Perron–Frobenius operator of T_r, we show, for β ∈ [0,1] and α ∈ (0, 1) , that the asymptotic behaviour of the iterates of P_r applied to observables with a singularity at β of order α is dependent on the structure of the ω-limit set of β with respect to T_r. The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.

Original language	English
Pages (from-to)	2585-2621
Number of pages	36
Journal	Ann. H. Poincare
Volume	17
Issue number	9
Early online date	19 Dec 2015
DOIs	https://doi.org/10.1007/s00023-015-0451-8
Publication status	Published - Sept 2016

Keywords

Banach Space
Ergodic Theory
Equlibrium states
Transfer Operator
Bounded Variation
Gibbs measures

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title = "On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps",

abstract = "We consider a family {Tr : [0, 1] → [0, 1] } r ∈ [0, 1] of Markov interval maps interpolating between the tent map T0 and the Farey map T1. Letting Pr denote the Perron–Frobenius operator of Tr, we show, for β ∈ [0,1] and α ∈ (0, 1) , that the asymptotic behaviour of the iterates of Pr applied to observables with a singularity at β of order α is dependent on the structure of the ω-limit set of β with respect to Tr. The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.",

keywords = "Banach Space, Ergodic Theory, Equlibrium states, Transfer Operator, Bounded Variation, Gibbs measures",

author = "Johannes Kautzsch and Marc Kesseb{\"o}hmer and Tony Samuel",

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doi = "10.1007/s00023-015-0451-8",

language = "English",

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journal = "Ann. H. Poincare",

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TY - JOUR

T1 - On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

AU - Kautzsch, Johannes

AU - Kesseböhmer, Marc

AU - Samuel, Tony

PY - 2016/9

Y1 - 2016/9

N2 - We consider a family {Tr : [0, 1] → [0, 1] } r ∈ [0, 1] of Markov interval maps interpolating between the tent map T0 and the Farey map T1. Letting Pr denote the Perron–Frobenius operator of Tr, we show, for β ∈ [0,1] and α ∈ (0, 1) , that the asymptotic behaviour of the iterates of Pr applied to observables with a singularity at β of order α is dependent on the structure of the ω-limit set of β with respect to Tr. The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.

AB - We consider a family {Tr : [0, 1] → [0, 1] } r ∈ [0, 1] of Markov interval maps interpolating between the tent map T0 and the Farey map T1. Letting Pr denote the Perron–Frobenius operator of Tr, we show, for β ∈ [0,1] and α ∈ (0, 1) , that the asymptotic behaviour of the iterates of Pr applied to observables with a singularity at β of order α is dependent on the structure of the ω-limit set of β with respect to Tr. The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.

KW - Banach Space

KW - Ergodic Theory

KW - Equlibrium states

KW - Transfer Operator

KW - Bounded Variation

KW - Gibbs measures

UR - https://arxiv.org/abs/1410.3805

U2 - 10.1007/s00023-015-0451-8

DO - 10.1007/s00023-015-0451-8

M3 - Article

SN - 1424-0637

VL - 17

SP - 2585

EP - 2621

JO - Ann. H. Poincare

JF - Ann. H. Poincare

IS - 9

ER -

On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

Abstract

Keywords

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