scholarly journals Spectral triples and Gibbs measures for expanding maps on Cantor sets

2012 ◽  
Vol 6 (4) ◽  
pp. 801-817 ◽  
Author(s):  
Richard Sharp
Keyword(s):  
Gibbs Measures ◽  
Cantor Sets ◽  
Expanding Maps ◽  
Spectral Triples

The singularity spectrum of the inverse of cookie-cutters

2009 ◽  
Vol 29 (4) ◽  
pp. 1075-1095 ◽  
Author(s):  
JULIEN BARRAL ◽  
STÉPHANE SEURET
Keyword(s):  
Gibbs Measures ◽  
Cantor Sets ◽  
Singularity Spectrum ◽  
Multifractal Formalism ◽  
Multifractal Measures

AbstractGibbs measures μ on cookie-cutter sets are the archetype of multifractal measures on Cantor sets. We compute the singularity spectrum of the inverse measure of μ. Such a measure is discrete (it is constituted only by Dirac masses), it satisfies a multifractal formalism and its Lq-spectrum possesses one point of non-differentiability. The results rely on heterogeneous ubiquity theorems.

CENTRAL LIMIT THEOREM FOR DIMENSION OF GIBBS MEASURES IN HYPERBOLIC DYNAMICS

2012 ◽  
Vol 12 (02) ◽  
pp. 1150019 ◽  
Author(s):  
RENAUD LEPLAIDEUR ◽  
BENOÎT SAUSSOL
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Equilibrium Measure ◽  
Gibbs Measures ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Axiom A ◽  
Surface Diffeomorphisms ◽  
The Central Limit Theorem

For an equilibrium measure of a Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. A noticeable consequence is that when this measure is not absolutely continuous, the probability that a ball of radius ε chosen at random have a measure smaller (or larger) than εδ is asymptotically equal to 1/2, where δ is the Hausdorff dimension of the measure. Our method applies to a class of non-conformal expanding maps on the d-dimensional torus. It also applies to conformal repellers and Axiom A surface diffeomorphisms and possibly to a class of one-dimensional non-uniformly expanding maps. These generalizations are presented at the end of the paper.

Symmetric Porosity and Symmetric Cantor Sets

1991 ◽  
Vol 17 (1) ◽  
pp. 19
Author(s):  
Evans
Keyword(s):  
Cantor Sets

scholarly journals Gauge transformations for twisted spectral triples

2018 ◽  
Vol 108 (12) ◽  
pp. 2589-2626 ◽  
Author(s):  
Giovanni Landi ◽  
Pierre Martinetti
Keyword(s):  
Spectral Triples ◽  
Gauge Transformations ◽  
Twisted Spectral Triples

scholarly journals On Gibbs Measures and Spectra of Ruelle Transfer Operators

2017 ◽  
Vol 60 (2) ◽  
pp. 411-421
Author(s):  
Luchezar Stoyanov
Keyword(s):  
Spectral Radius ◽  
Spectral Properties ◽  
Gibbs Measures ◽  
Transfer Operator ◽  
Frobenius Theorem ◽  
Transfer Operators

AbstractWe prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

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