Large deviations and multifractal analysis for expanding countably-branched Markov maps

• Main Author: Dungca, Jason Tomas
• Other Authors: Jordan, Thomas
• Published: University of Bristol 2018
• Subjects: 510
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.761180

We will use the theory of thermodynamic formalism for countable Markov shifts to pose and solve problems in multifractal analysis and large deviations. We start with an introduction outlining results in thermodynamic formalism, multifractal analysis, and large deviations in Chapter 1. We state necessary concepts and results from dynamical systems, ergodic theory, thermodynamic formalism, dimension theory, and large deviations in Chapter 2. In Chapter 3, we consider the multifractal analysis for Gibbs measures for expanding, countably branched Markov maps. We will find conditions for the multifractal spectrum to have various numbers of phase transitions. Finally, in Chapter 4, we consider an expanding, countably-branched Markov map T, the countable Markov shift, and a locally Hölder potential f. The behaviour of the dynamical system (T(lambda),(0,1]) depends on the value of lambda. We will aim to form a large deviation principle for f for a fixed lambda in (1/2,1) and we will discuss the method for forming a such a principle for a lambda in (0,1/2) in Chapter 4’s introduction.