ONE DAY MEETING: PIECEWISE ISOMETRIES

This is a London Dynamical Systems Group LDSG workshop. LDSG is a joint research network of the dynamical systems groups at Imperial College London, Queen Mary, University of London, University College London and the University of Surrey, supported by a Scheme 3 grant from the London Mathematical Society.

Proposed programme

• 12.15 - 1.00pm Franco Vivaldi (Queen Mary, University of London)

  Interval exchange maps over algebraic number fields
  

  Abstract: We consider the restriction of interval-exchange maps to algebraic number fields, which leads to dynamics on lattices. We show that if the map is uniquely ergodic and renormalizable, then the scaling constant is a unit in the ring of integers associated with the scaling process. We study in detail a specific example, the cubic Arnoux-Yoccoz map, for which the inverse of the scaling constant has the Pisot property. We prove that each invariant lattice decomposes nontrivially into the union of finitely many orbits. The renormalization dynamics is described in terms of an expansion in a Pisot base, where the digits are algebraic integers.

• 2.30 - 3.15pm Orla Feely (University College, Dublin)

  Piecewise Isometries in Electronic Circuits and Systems
  

  Abstract: Many widely-used electronic circuits are modelled by nonlinear maps. Because of the nature of the nonlinearities in these circuits these maps are often discontinuous, piecewise-linear and area-preserving. This talk will present a short survey of such maps as they arise in electronic circuits and systems.

• 3.15 - 4.00pm Marcello Trovati (Exeter University)

  Survey of tangency properties in disc packings generated by some PWIs
  

  Abstract: Disc packings associated to PWIs have interesting tangency properties. In particular the talk will focus on two particular examples, the Arbelos packing and a PWI acting on an isosceles triangle introduced by Goetz (1998). These two examples generate interesting and complex packings and we believe that their tangency properties could give a useful insight into more general dynamical properties of piecewise isometric systems as well as other topics seemingly unrelated to PWIs.

• 4.45 - 5.30pm Arek Goetz (San Francisco State University)

  Recurrence of points in the two half plane map.
  

  Abstract: We will report on our joint work with Anthony Quas (Victoria) about the dynamics of the map T defined on the complex plane. The maps acts as rotation on the lower and upper halfplane and it is Lebesgue measure preserving. We show that every neighborhood of infinity contains periodic points. Also, despite the lack of an obvious finite measure preserved by T, we still have the Poincare Recurrence Property, that is almost all points come back near where they start. This recent work closes an open case studied a few years ago with Michael Boshernitzan.

  Time permitting we will discuss some observations on symmetry breaking under taking first return maps.

Arrangements for the day

The talks will take place in room G2, ground floor of the mathematics building, QMUL.

Tea and coffee will be available at 11:45 and 4:00.

Lunch will be between 1pm and 2pm. There will also be an evening meal in a local restaurant.

Directions to the University, including a location map.

Previous meetings

For details of previous meetings, click here or here.

Organisers

Henk Bruin/Jonathan Deane
Department of Mathematics
University of Surrey
Guildford
Surrey GU2 7XH
Tel: 01483 689253/689104
email: H.Bruin@surrey.ac.uk / J.Deane@surrey.ac.uk

Franco Vivaldi
Department of Mathematics
Queen Mary, University of London
Mile End Road
London E1 4NS
Tel: 020 7882 5488
e-mail: F.Vivaldi@qmul.ac.uk