Banff workshop  08w5046

Low Complexity Dynamics

Provisional schedule for 25th May- 30th May 2008

https://www.birs.ca/birs-rsvp.php




Sunday Monday Tuesday Wednesday Thursday Friday

25th May 2008 26th May 2008 27th May 2008 28th May 2008 29th May 2008 30th May 2008


BREAKFAST BREAKFAST BREAKFAST BREAKFAST BREAKFAST
8.45-9.00

Introduction from BIRS station manager, Brenda Williams




9.00-9.45
Rich Schwarz,
Unbounded orbits for outer billiards
Michael Baake,
A glimpse at aperiodic order
Bob Devaney,
Cantor and Sierpinski, Julia and Fatou: crazy topology in complex dynamics
David Damanik,
Spectral theory of Schr"odinger operators with dynamically generated potentials.
Anthony Quas,
TBA
9.45-10.30
Yitwah Cheung,
Best rational approximations
Rod Edwards,
Dynamics of glass networks
Henk Bruin,
Most piecewise contractions are asymptotically periodic
Chai-Wah Wu,
Digital halftoning and sensor placement
Informal discussions

10.30-11.00
COFFEE COFFEE COFFEE COFFEE COFFEE
11.00-11.45
Ilie Ugarcovici,
Continued fractions, natural extension maps and their geometric structure
John Roberts,
The dynamics of reversible rational maps over finite fields via a combinatoric model using random involutions.
Pat Hooper,
Some irrational polygons have many periodic billiard paths
Tomasz Nowicki, Convex dynamics with constant input: The case of the acute triangle. Informal discussions
11.45-12.15
Nicolas Bedaride,
Symbolic dynamics of the dual billiard map
Sinan Gunturk,
Simultaneous beta-representations
Rob Sturman
Piecewise isometries and mixing in granular tumblers
Simon Lloyd,
On closing bounded type recurrence on the torus
Informal discussions
12.30-2.00
LUNCH
1.00-2.00 optional tour of Banff centre followed by
2.00 Group photograph in front of Corbett Hall
LUNCH LUNCH LUNCH LUNCH
2.00-2.45
(starts 5 mins late)
Sergei Tabachnikov,
Polygonal outer billiards
Franco Vivaldi,
Interval exchange maps over algebraic number fields
OUTING John Lowenstein,
Computational explorations relevant to Ashwin's conjecture for digital-filter maps
DEPART
2.45-3.15

(starts 5 mins late)
Benito Pires,
On Cr closing for flows on orientable and nonorientable 2-manifolds
Peter Ashwin,
Packings from piecewise isometries

Congping Lin,
Some Properties of Piecewise Parabolic Maps

3.15-4.00

TEA
TEA

TEA

4.00-4.45

Karl Petersen,
Some zero-dimensional dynamical systems generated by reinforced walks on graphs
Sebastian Ferenczi,
Languages and inductions for interval-exchange transformations

Arek Goetz,
A symbolic package for exploration of rational piecewise isometries

4.45-5.30 ARRIVAL
Michael Boshernitzan,
TBA
Matt Nicol,
TBA

Denis Giadashev,
Renormalization in area-preserving dynamics

6.00-7.30 DINNER DINNER DINNER
DINNER
8.30-9.30 Cheese and wine

PROBLEM SESSION
Bring your favourite open problems to present to the group!




Last edit: 12/5/08 (PA)


Talk abstracts (where available)



Peter Ashwin

Packings from piecewise isometries

I will discuss some results and open problems concerning packings induced by periodically coded points for planar piecewise isometries.

Michael Baake

A glimpse at aperiodic order

The discovery of solids with long-range order and non-crystallographic symmetries some 25 years ago has sparked the theory of aperiodic order, also building on various mathematical predecessors in harmonic analysis and discrete geometry. This talk is intended to give an overview of the field, with focus on the various connections to the traditional diciplines, including dynamical systems of zero entropy.

Nicolas Bedaride

Symbolic dynamics of the dual billiard map

We consider the dual billiard map inside a polygon. We code the orbit of a point with the natural partition associated to this map. We will describe the language of admissible words for this map for some regular polygons as the regular pentagon.

Michael Boshernitzan

TBA

Henk Bruin

Most piecewise contractions are asymptotically periodic.

For piecewise isometries low-complexity chaos is caused by discontinuities. For piecewise contractions, the discontinuities can still cause the system to be chaotic, but typically the only
asymptotic dynamics is periodic. In this talk I want to report on a joint paper with Jonathan Deane, which proves this for piecewise contractions of the plane.
polygons as the regular pentagon.

Yitwah Cheing

Best (Rational) Approximations

Given x in R^d, a "best (rational) approximation" to x is roughly defined to be any rational that is closer to x than any other rational of smaller height.  Recall that the height of a rational is the least common multiple of the denominators of its coordinates, each in lowest terms.  The sequence of best approximations to a given x are naturally ordered according to height.  In the case d=1, they coincide with the sequence of convergents in the simple continued fraction of x.  In this talk, we present a dynamical interpretation of the sequence of best approximations in terms of a certain flow on the space of unimodular lattices in R^{d+1}. Using this, we develop analogs of some of the well-known facts about convergents.  If time permits, we will also mention some  applications to the study of singular vectors.

David Damanik

Spectral theory of Schr"odinger operators with dynamically generated potentials

TBA

Bob Devaney

Cantor and Sierpinski, Julia and Fatou: Crazy Topology in Complex Dynamics

In this talk, we shall describe some of the rich topological structures that arise as Julia sets of certain complex functions including the exponential and rational maps. These objects include Cantor bouquets, indecomposable continua, and Sierpinski curves.

Rod Edwards

Dynamics of Glass Networks

Abstracted models of gene regulatory networks take the form of piecewise-linear differential equations called 'Glass Networks'. Their Poincare maps are also piecewise-fractional linear, each piece corresponding to a cycle in the state transition graph, for which there is thus a natural symbolic dynamics. I will give some results relating the network structure (graph topology) to the Poincare maps and show how maps with interesting dynamical properties can arise.
Sebastien Ferenczi

Languages and inductions for interval-exchange transformations

We give a combinatorial characterization of the low complexity infinite words which are natural codings of interval exchange transformations, and in the best cases generate them explicitely through a self-dual induction process

Denis Giadashev

Renormalization in area-preserving dynamics

Arek Goetz

A symbolic package for exploration of rational piecewise isometries

Sinan Gunturk

Simultaneous beta-representations

TBA

Patrick Hooper

Some irrational polygons have many periodic billiard paths

A polygon is rational if all of its angles are irrational multiple of pi and irrational otherwise. Much more is known about periodic billiard paths in rational polygons than periodic billiard paths in irrational polygons. We will apply some ideas from rational billiards to show that there are rational polygons where the number of periodic billiard paths of length less than t grows superlinearly in t.

Congping Lin

Some Properties of Piecewise Parabolic Maps
Abstract: Because the inverse of 2-torus invertible parabolic maps may not be parabolic, we need to study planar piecewise parabolic maps (PWP). This talk will discuss some properties of such PWPs.

Simon Lloyd

On closing bounded type recurrence on the torus

Consider a smooth vector field with a recurrent point p. The open C^r Closing Lemma problem, r>1, asks whether p can be made periodic by a C^r-small perturbation of the vector field. For vector fields on the torus, this is known to be possible for vector fields exhibiting 'unbounded type' recurrence. Here we examine the open, complementary case, and by means of a wandering interval argument, we obtain a positive answer to the C^r Closing Lemma problem for a class of vector fields of bounded type.

 John Lowenstein

Computational explorations relevant to Ashwin's conjecture for digital-filter maps

P. Ashwin has conjectured that a family of  2D piecewise isometries, of interest in the theory of digital filters, is characterized by the orbits in the exceptional set (complementary to the elliptic islands) having non-zero Lebesgue measure.  We review the evidence for this intriguing possibility, and describe some recent computational explorations of certain promising regions of phase space.
This work is a collaboration with F. Vivaldi (Queen Mary University of London).

Matt Nicol

TBA

Tomasz Nowicki

Convex dynamics with constant input: The case of the acute triangle.

While investigating on-line algorithms in AD conversion certain piecewise linear  maps can be  defined in terms of a convex polytope. When the convex polytope is a simplex, the resulting map has a duanature. On one hand it is defined on R^d and acts as a piecewise translation. On the other it can be viewed as a translation on the d-torus. What relates its two roles? A natural answer would be that there exists an invariant fundamental region into which all orbits under piecewise translation eventually enter. We prove this for d=1 and for acute and right triangles--i.e non-obtuse triangles. The case of obtuse triangles and higher dimensional simplices is still an open problem.
Karl Peterson

Some zero-dimensional dynamical systems generated by reinforced walks on graphs.

Possible abstract: In joint work with Sarah Bailey Frick, we have been setting up and studying adic (Bratteli-Vershik) systems determined by reinforced walks on graphs. Whenever an edge is traversed, new edges are added to the graph according to a fixed rule. The possible histories of a walker then determine a Bratteli diagram and an associated transformation defined on (most of) a Cantor set. Specific examples include the Pascal, Euler, reverse Euler, and Stirling systems.  Determination of  the dynamical properties of these systems  (partially accomplished)  has  connections with interesting facts and problems in combinatorics, probability, and number theory, which we discuss.
Benito Pires

On Cr closing for flows on orientable and nonorientable 2-manifolds

The Cr closing lemma for flows can be stated in the following way: Given a Cr vector field X defined on a compact surface M and a non-trivial recurrent point p of X, does there exist a vector field Y arbitrarily Cr close to X having a periodic trajectory passing through p? In other words, can a non-trivial recurrent trajectory be closed by an arbitrarily small Cr-perturbation of the original vector field? C. Pugh gave a positive answer to this problem in topology C1 (for flows and diffeomorphisms on n-manifolds) whereas C. Gutierrez and C. Carroll presented partial, positive results in the topology Cr, r>1, for flows defined on orientable surfaces. In this talk I shall present a partial, positive answer for this problem in topology Cr, r>1, for a class of flows defined on every compact surface (orientable and nonorientable) supporting non-trivial recurrence. Joint work with C. Gutierrez.

A Quas

TBA

John Roberts

The dynamics of reversible rational maps over finite fields via a combinatoric model using random involutions.

Reversible rational maps are those maps in d-dimensional space that can be written as the composition of 2 rational involutions. Most generally, the denominators of the rational expressions can vanish, producing singularities.  We study the reduction of such rational maps to finite fields and look to study the proportion of the finite phase space occupied by cycles and by aperiodic orbits and the length distributions of such orbits. We find that the dynamics of these low-complexity highly deterministic maps has some universal (i.e. map-independent) aspects.  The dynamics is well explained using averages over an ensemble of pairs of random involutions in the finite phase space.

Rich Schwarz

Unbounded orbits for outer billiards

I will explain my recent solution of the Moser-Neumann problem - the question about unbounded orbits in outer billiards.  The main theorem is that outer billiards on any irrational kite has uncountably many unbounded orbits.  An irrational kite is a kite-shaped quadrilateral that is not affine equivalent to a lattice polygon.  One key idea in my proof is that outer billiards in the plane compactifies, in a certain sense, into a higher dimensional polytope exchange map. I will illustrate all my results with Billiard King, a graphical user interface I built for the purpose of investigating this topic.

Rob Sturman

Piecewise isometries and mixing in granular tumblers

TBA

Serge Tabachnikov

On complexity of polygonal outer billiards

TBA

Ilie Ugarcovici

Continued fractions, natural extension maps and their geometric structure

We describe new types of continued fractions and geometric realizations of their natural extension maps. The domain of such a two-dimensional extension map coincides with a canonically associated attracting set. We analyze the geometric structure of these attracting sets and derive some ergodic properties of the continued fraction maps. This is joint work with S. Katok and D. Zagier.

Franco Vivaldi

Interval-exchange maps over finite number fields

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.

Chai-Wah Wu

Digital halftoning and sensor placement

One engineering problem where low complexity dynamics has found great applicability is digital halftoning.  In this talk I look at some relationships between the digital halftoning problem and the sensor placement problem in sensor networks in the hope of fostering collaboration among researchers in these areas.