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.
Organisations
People |
ORCID iD |
| Mark Pollicott (Principal Investigator) |
Publications
Aimino R
(2021)
Thermodynamic Formalism - CIRM Jean-Morlet Chair, Fall 2019
Baker S
(2020)
Equidistribution results for sequences of polynomials
in Journal of Number Theory
Baker S
(2020)
On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne
in Advances in Mathematics
Baker S
(2019)
Exceptional digit frequencies and expansions in non-integer bases
in Monatshefte für Mathematik
Baker S
(2020)
Two bifurcation sets arising from the beta transformation with a hole at 0
in Indagationes Mathematicae
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)
Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms
in Linear Algebra and its Applications
Morris I
(2010)
A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory
in Advances in Mathematics
Morris I
(2013)
Mather sets for sequences of matrices and applications to the study of joint spectral radii
in Proceedings of the London Mathematical Society
MORRIS I
(2009)
The Mañé-Conze-Guivarc'h lemma for intermittent maps of the circle
in Ergodic Theory and Dynamical Systems
Morris I
(2012)
The generalised Berger-Wang formula and the spectral radius of linear cocycles
in Journal of Functional Analysis
Morris Ian D.
(2009)
The generalized Berger-Wang formula and the spectral radius of linear cocycles
in arXiv e-prints
Pollicott M
(2023)
Accurate Bounds on Lyapunov Exponents for Expanding Maps of the Interval.
in Communications in mathematical physics
Pollicott M
(2022)
Explicit examples of resonances for Anosov maps of the torus
in Nonlinearity
Pollicott M
(2021)
The Schottky-Klein prime function and counting functions for Fenchel double crosses
in Monatshefte für Mathematik
Pollicott M
(2021)
Thermodynamic Formalism - CIRM Jean-Morlet Chair, Fall 2019
Pollicott M
(2021)
Fourier multipliers and transfer operators
in Journal of Fractal Geometry
Pollicott M
(2021)
Effective estimates of Lyapunov exponents for random products of positive matrices
in Nonlinearity
Pollicott M
(2020)
Exact dimensional for Bernoulli measures and the Gauss map
in Proceedings of the American Mathematical Society
Pollicott M
(2019)
Geometric and Ergodic Aspects of Group Actions
| Description | 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 |