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Thermodynamic formalism and dimension gaps
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Jurga, Natalia Anna (2018) Thermodynamic formalism and dimension gaps. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3228221~S15
Abstract
Given an expanding Markov map T : [0; 1] → [0; 1] which admits an absolutely continuous invariant probability measure, we say that T gives rise to a dimension gap if there exists some c > 0 for which supp dim µp 1 . c, where µp denotes the Bernoulli measure associated to the probability vector p. We prove that under a `non-linearity condition' on T, there is a dimension gap.
Our approach differs considerably to the approach of Kifer, Peres and Weiss in [KPW], who proved a similar result. The first part of our proof involves obtaining uniform lower estimates on the asymptotic variance of a class of potentials. Tools from the thermodynamic formalism of the countable shift play a key role in this part of the proof. The second part of our proof revolves around a `mass redistribution' technique.
We also study a class of `Käenmäki measures' which are supported on self- affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane. We prove that such a measure is exact-dimensional and that its dimension satisfies a Ledrappier-Young formula. This is similar to the recent results of Bárány and Käenmäki [BK], who proved an analogous result for quasi- Bernoulli measures. While the measures we consider are not quasi-Bernoulli, which takes us out of the scope of [BK], we show that the measures can be written in terms of two quasi-Bernoulli measures on an associated subshift and use this to prove the result.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Thermodynamics, Mappings (Mathematics), Invariants | ||||
Official Date: | July 2018 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Pollicott, Mark | ||||
Extent: | vi, 174 leaves | ||||
Language: | eng |
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