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Title: Thermodynamic formalism and dimension gaps
Author: Jurga, Natalia Anna
ISNI:       0000 0004 7431 7174
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2018
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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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics