Dissertation

Gibbs States in Statistical Mechanics and Dynamical Systems

This dissertation is organised into three parts, each addressing a distinct research area related to Gibbs measures.

Author

M. Makhmudov

Date

02 September 2025

Links

Thesis in Leiden Repository

Part I examines the various definitions of Gibbsianity found in the literature on Statistical Mechanics and Dynamical Systems. Chapter 2 compares these definitions in terms of their generality, while Chapter 3 investigates the Gibbs properties of equilibrium states in a broad setting, identifying a minimal condition under which such states are Gibbs. 

Part II shifts focus to one-dimensional systems, comparing—through the lens of absolute continuity—whole-line and half-line Gibbs measures associated with a given potential. This analysis is connected to the study of principal eigenfunctions of the Ruelle–Perron–Frobenius transfer operators corresponding to the underlying potential. In particular, this part initiates a systematic investigation of transfer operators for long-range potentials. 

Part III explores the multifractal properties of systems governed by large deviations. Motivated by the longstanding parallels between multifractal formalism and large deviations theory, it takes initial steps toward establishing a systematic framework for deriving multifractal formalism directly from large deviations principles.

• eigenfunctions
• equilibrium states
• gibbs states
• large deviations
• multifractal spectrum
• transfer operators
• whole-line vs half-line gibbsianness