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A one day study in Ergodic Theory and Dynamical Systems

22nd October 2015, CEMPS – University of Exeter, Organiser: Ana Rodrigues

Morning session (in Harrison Building, room  HAR 171)
  • 11:30 – 12:30 Paulo Varandas (UFBA, Brasil)
  • 12:30 – 13:30 Thomas Jordan (Bristol, UK)
Afternoon session (in Laver Building, room Laver 212)
  • 15:30 – 16:30 Peter Ashwin (Exeter, UK)
  • 16:30 – 17:30 Yiwei Zhang (Exeter, UK)
Paulo Varandas – Topological methods for problems in ergodic theory

Many chaotic dynamical systems are ‘rich’ from the topological viewpoint. This richness can be expressed by the fact that pseudo-orbits are shadowed by true orbits for the dynamics (shadowing property) or that pieces of orbits are well approximated by a orbit (specification property). These properties are satisfied by uniformly hyperbolic dynamics and some weaker versions have been established for wide classes of non-uniformly hyperbolic dynamical systems. In this talk we shall recall some of these weaker notions of shadowing and specification and describe how these can be used as very usefull tools in some problems arising in ergodic theory.

Thomas Jordan – Minkowski’s question mark function

Minkowski’s question mark function is a homeomprphism from [0,1] to itself which maps quadratic irrationals into the rationals. We’ll show how to view this function as a conjugacy between dynamical systems and as an invariant measure for the Gauss map (x\to 1/x\mod 1). We’ll then outline how a combination of large deviations and methods from a paper of Kaufman can be used to show the Fourier-Stieltjes transform of this function decays polynomially and give a few consequences of this fact. (Joint work with Tuomas Sahlsten).

Peter Ashwin  – Piecewise isometric dynamics; some puzzles and problems

I will introduce and discuss some aspects of the dynamics of piecewise isometries (PWIs). These are iterated maps that are geometrically very simple (just isometries) on a number of “atoms” – convex regions that form a partition of phase space. They can be seen generalizations of interval exchange transformations to higher dimensions. PWis may arise in a number of applications but are hard to understand using standard dynamical tools based around hyperbolicity. I will outline a few results and some unsolved problems related to the dynamics, geometry and measure of the various invariant subsets that arise for PWIs in two dimensions.

Yiwei Zhang – A glimpse of thermodynamics formalism for interval maps

I will take a short survey on the basic setting of thermodynamics formalism for interval maps. In particular, I will introduce the standard transfer operator method in the uniformly hyperbolic maps setting, and explain how the spectral structure of this operator links with the existence/uniqueness of equilibrium state and its exponential decay of correlation. I will also explain how to extend this method for the non-uniformly hyperbolic interval maps.

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