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One-day study in ergodic theory and dynamical systems

We are pleased to announce the second one-day study in Ergodic Theory and Dynamical Systems to be held at the University of Exeter on 17th November 2017. If you have any enquiries please ask .

Programme:

11:30-12:30 Jimmy Tseng (Exeter University)

Title: An introduction to homogeneous dynamics and its interactions with number theory

Room: Peter Chalk 1.1

12:30-13:30 Pedro Peres (Exeter University)

Title: Embeddings of Interval Exchange Transformations into Planar Piecewise Isometries

Room: Peter Chalk 1.1

13:30- 15:30 Lunch

15:30-16:30 Samuel Roth (Opava University, CZ)

Title: Constant Slope Models and Perturbation

Room: Harrison 254

16:30-17:30 Zuzana Roth  (Opava University, CZ)

Title: Li-Yorke sensitivity and a conjecture of Akin and Kolyada

Room: Harrison 254

 

Speakers and abstracts:

Jimmy Tseng (Exeter University)

An introduction to homogeneous dynamics and its interactions with number theory

I will introduce the study of homogeneous dynamics through a particular homogeneous space, namely the space of unimodular lattices, and a particular flow, namely a suitable diagonal flow.  I will show how the dynamics corresponds to Diophantine approximation, which is a subfield of number theory concerned with how well real objects (numbers, vectors, matrices) are approximated by suitable rational ones.

We will discuss ergodic properties of the diagonal flow and its consequences for number theory.  In particular, I will mention a recent paper of mine, joint with Athreya and Parrish, which gives us a conceptually simple way of showing a classical theorem in Diophantine approximation.

Pedro Peres (Exeter University)

Embeddings of Interval Exchange Transformations into Planar Piecewise Isometries

Although piecewise isometries (PWIs) are higher dimensional generalizations of one dimensional interval exchange transformations (IETs), their generic dynamical properties seem to be quite different. In this talk I will consider embeddings of IET dynamics into PWI with a view to better understanding their similarities and differences.  I will introduce a family of PWIs with apparent abundance of invariant nonsmooth fractal curves supporting IETs. I will recall some notions used in the study of IETs, introduce some new tools and describe how they can be used to establish some results on the existence of embeddings of IETs into PWIs. (Joint work with P. Ashwin, A. Goetz and A. Rodrigues).

Samuel Roth (Opava University, CZ)

Constant Slope Models and Perturbation

We work in the space of transitive, piecewise monotone maps of a fixed modality m with the topology of uniform convergence. There is an operator on this space which assigns to a map its constant slope model. This operator is discontinuous at points (maps) where perturbation can lead to a jump in entropy. Alseda and Misiurewicz conjectured that these are the only discontinuity points. We confirm the conjecture by a technique of “counting preimages.” (joint work with Michal Malek)

Zuzana Roth  (Opava University, CZ)

Li-Yorke sensitivity and a conjecture of Akin and Kolyada

Akin and Kolyada in 2003 [1] conjectured that every minimal system with a weak mixing factor, is Li-Yorke sensitive. Our interest in this problem comes from a survey paper by Li and Ye [2] which came out last year. This talk will deal with a proof of the conjecture for minimal 2-point extensions of weak mixing systems,  which became an inspiration for a more complex solution by Mlichova [3]. The talk will be closed with results that showed up in the last year. (joint work with Samuel Roth)

Bibliography:

[1] Ethan Akin and Sergii Kolyada. Li-Yorke sensitivity. Nonlinearity , 16(4):1421, 2003.

[2] Jian Li and Xiangdong Ye. Recent development of chaos theory in topological dynamics. Acta Mathematica Sinica, English Series}, 32(1):83–114, 2016.

[3] Michaela Mlichova. Li-Yorke sensitive and weak mixing dynamical systems. Journal of Difference Equations and Applications, 0(0):1–8, 0.


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