Day of Dynamics -- Dynamical Systems

DAY OF DYNAMICS 

Thanks to the International Visitor Program we are getting an influx of diverse
expertise in the field of dynamical systems this month.  You are all invited on Mon Feb
18th for a day of dynamical systems talks from both the pure and applied side on some of
the emerging topics in the respective areas.  

This is an opportunity to foster interdisciplinary collaboration especially amongst
early-career researchers and students.  

Time: Mon Feb 18th 10:30 - 5:15 
Place: USYD Carslaw 373 

Speakers: 

Nalini Joshi (USYD) 
Colin Guillarmou (CNRS, Paris Sud) 
Marco Mazzucchelli (CNRS, ENS Lyon) 
Nobutaka Nakazono (Aoyama Gakuin University) 
Milena Rodnovic (USYD) 
Alexander Fish (USYD) 
Giorgio Gubbiotti (USYD) 

Program: 

10:30-11:15 Marco Mazzucchelli (CNRS, ENS Lyon) 
Min-Max Characterization of Zoll
Riemannian Manifolds.  

11:15 - 12:00 Milena Radnovic (USYD) 
Billiards within quadrics and extremal polynomials 

12:00-1:00 LUNCH 

1:00 - 1:45 Nalini Joshi (USYD) 
Dynamics of a Painlevé equation in initial-value space 

1:50-2:35 Colin Guillarmou (CNRS, Paris Sud) 
The marked length spectrum in negative
curvature 

BREAK 

2:50- 3:35 Alexander Fish (USYD) 
Twisted recurrence via polynomial walks 

3:40- 4:25 Nobutaka Nakazono (Aoyama Gakuin University) 
Affine Weyl group symmetry of
the discrete power function 

4:30 - 5:15 Giorgio Gubbiotti (USYD)
 Integrable discrete autonomous quad-equations
admitting, as generalized symmetries, known five-point differential-difference equations 

Abstracts (ordered according to schedule): 

Marco Mazzucchelli 
Min-Max Characterization of Zoll Riemannian Manifolds 

A closed Riemannian manifold is called Zoll when its unit-speed geodesics are all
periodic with the same minimal period.  This class of manifolds has been thoroughly
studied since the seminal work of Zoll, Bott, Samelson, Berger, and many other authors.
It is conjectured that, on certain closed manifolds, a Riemannian metric is Zoll if and
only if its unit-speed periodic geodesics all have the same minimal period.  In this
talk, I will first discuss the proof of this conjecture for the 2-sphere, which builds
on the work of Lusternik and Schnirelmann.  I will then show an analogous result for
certain higher dimensional closed manifolds, including spheres, complex and quaternionic
projective spaces: a Riemannian manifold is Zoll if and only if two suitable min-max
values in a free loop space coincide.  This is based on joint work with Stefan Suhr.  

Milena Radnovic 
Billiards within quadrics and extremal polynomials 

We will present a comprehensive study of periodic trajectories of the billiards within
ellipsoids in d-dimensional space, based on a relationship established between such
trajectories and the extremal polynomials on a systems of d intervals on the real line.
We will also give a case study of trajectories of small periods T, d ≤ T ≤ 2d.  

Nalini Joshi 
Dynamics of a Painlevé equation in initial-value space 

Painlevé equations are nonlinear nonautonomous second-order ODEs, which appear as
universal models in physics.  Very little explicit information is known about their
transcendental solutions.  We describe a dynamical approach to deduce their global
behaviors in a compactified, regularized space of initial values, which share geometric
properties with elliptic surfaces.  Such a space was first described by Okamoto (1979).
We consider a construction suitable for an asymptotic limit, and prove properties about
the complex limit sets of solutions.  Joint work with Duistermaat (2011), Howes (2014)
and Radnovic (2015-19).  

Colin Gullarmou 
The marked length spectrum in negative curvature 

We review the problem asked by Burns and Katok about determining a Riemannian metric on
a closed negative manifold from the length of its closed geodesics.  

Alexander Fish 
Twisted recurrence via polynomial walks 

We will show how polynomial walks can be used to establish a twisted recurrence for sets
of positive density in Z^d.  In particular, we will demonstrate that if Γ ≤ GL_d(Z)
is finitely generated by unipotents and acts irreducibly on R^d, then for any set B ⊂
Z^d of positive density, there exists k ≥ 1 such that for any v ∈ kZ^d one can find
γ ∈ Γ with γv ∈ B − B.  Also we will show a non-linear analog of Bogolubov’s
theorem – for any set B ⊂ Z^2 of positive density, and p(n) ∈ Z[n], p(0) = 0, deg
p ≥ 2, there exists k ≥ 1 such that kZ ⊂ {x + p(y) | (x, y) ∈ B − B}.  Joint
work with Kamil Bulinski.  

Nobutaka Nakazono 
Affine Weyl group symmetry of the discrete power function 

In this talk, we show that the discrete power function associated with circle patterns
of Schramm type can be obtained from a space-filling cubic lattice, each cube has CAC
property, and its affine Weyl group symmetry.  Moreover, we show that this cubic lattice
and its symmetry are derived form the affine Weyl group symmetry of the sixth Painlev\’e
equation.  This work has been done in collaboration with Profs Nalini Joshi, Kenji
Kajiwara, Tetsu Masuda and Dr Yang Shi.  

Giorgio Gubbiotti 
Integrable discrete autonomous quad-equations admitting, as
generalized symmetries, known five-point differential-difference equations 

We construct the autonomous quad-equations which admit as symmetries the five-point
differential-difference equations belonging to known lists found by Garifullin, Yamilov
and Levi.  The obtained equations are classified up to autonomous point transformations
and some simple non-autonomous transformations.  We discuss our results in the framework
of the known literature.  There are among them a few new examples of both sine-Gordon
and Liouville type equations.