# Maximizing measures in hyperbolic dynamics

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

In the study of dynamical systems, the long term asymptotic behaviour is often best understood in terms of properties of invariant measures. In the particular setting of Lagrangian flows, Mane and Mather formulated a number of important questions about those measures whose integrals maximized a particular integral. As part of a general theory, one can ask similar questions about quite diverse dynamical systems. In the particular case of hyperbolic (or chaotic) dynamical systems, we can consider a natural class of measures called Gibbs measures, whose study originated in the mathematical theory of Statistical Mechanics, and the work of Ruelle and Sinai. This proposal relates to how maximizing measures for typical functions for these hyperbolic systems can be approximated by these better behaved Gibbs measures. This has important implications for understanding the quite complicated, but important, maximizing measures in terms of much simpler Gibbs measures.

## People |
## ORCID iD |

Mark Pollicott (Principal Investigator) |

### Publications

Aimino R
(2021)

*Thermodynamic Formalism - CIRM Jean-Morlet Chair, Fall 2019*
Colognese P
(2020)

*Dynamics: Topology and Numbers*
Hare K
(2011)

*An explicit counterexample to the Lagarias-Wang finiteness conjecture*in Advances in Mathematics
Jenkinson O
(2021)

*How Many Inflections are There in the Lyapunov Spectrum?*in Communications in Mathematical Physics
Jenkinson O
(2020)

*Dynamics: Topology and Numbers*
Kleptsyn V
(2022)

*Uniform lower bounds on the dimension of Bernoulli convolutions*in Advances in Mathematics
Morris I
(2010)

*A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory*in Advances in Mathematics
Morris I
(2012)

*The generalised Berger-Wang formula and the spectral radius of linear cocycles*in Journal of Functional Analysis
MORRIS I
(2009)

*The Mañé-Conze-Guivarc'h lemma for intermittent maps of the circle*in Ergodic Theory and Dynamical Systems
Morris I
(2010)

*Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms*in Linear Algebra and its ApplicationsDescription | The grant was successful in making substantial progress on properties of maximizing measures. This stimulated research in related areas by other researchers, with connections to analysis, geometry, dynamical systems, |

Exploitation Route | The RA went on to told a lectureship in Surrey. The work on the grant has stimulated research by many other authors. |

Sectors | Digital/Communication/Information Technologies (including Software),Education,Transport |

URL | http://www.surrey.ac.uk/maths/people/ian_morris/#publications |