Abstracts
Plenary speaker
Amie Wilkinson (University of Chicago)
Dynamical asymmetry is C^1-typical
I will discuss a result with Bonatti and Crovisier from 2009 showing that the C^1 generic diffeomorphism f of a closed manifold has trivial centralizer; i.e. fg = gf implies that g is a power of f. I’ll discuss features of the C^1 topology that enable our proof (the analogous statement is open in general in the C^r topology, for r>1). I’ll also discuss some features of the proof and some recent work, joint with Danijela Damjanovic and Disheng Xu that attempts to tackle the non-generic case.
Invited speakers
Does Ricci flow preserve the Anosov property?
I will describe a joint work in progress with Keith Burns and Dong Chen. We construct a surface with Anosov geodesic flow which acquires conjugate points when it evolves under Ricci flow. Since metrics with Anosov geodesic flow have no conjugate points, this shows that the Anosov property is not preserved by Ricci flow. This gives a negative answer to a question of Manning from 2004.
Combinatorics of Quadratic Rational Maps with a critical n-cycle
You may have heard of the Mandelbrot set, a combinatorial fractal object which lives inside the parameter space of quadratic polynomials {z^2+c}. Why does the Mandelbrot set emerge within this moduli space? What happens if instead of looking at quadratic polynomials, which are Quadratic Rational Maps with a fixed critical point at infinity, we looked at the QRMs with a critical point in an n-cycle? These are the so-called Per_n(0) curves, and they turn out to admit a rich combinatorial structure at least in part illuminated by the structure of the Mandelbrot set. In this talk, we’ll present an array of recent progress detailing these curves and how to think about them, as well as some future directions and open problems.
Eigenvalues and the stabilized automorphism group of minimal systems
The stabilized automorphism group of a dynamical system (X,T) is the group of all self-homeomorphisms of X that commute with some power of T. In this talk, we will describe the stabilized automorphism group of minimal. systems. The main result that we will prove is that if two minimal systems have isomorphic stabilized automorphism groups and each has at least one non-trivial rational eigenvalue, then the systems have the same rational eigenvalues.
Hidden Symmetry Rigidity of Geodesic Flows
An old result of Bochner proves that closed Riemannian manifolds of negative Ricci curvature admit only finitely many isometries. On the other hand, work beginning with Eberlein, and later extended by Farb and Weinberger, shows that rigidity in the presence of too many isometries still occurs provided one looks at covers of a closed manifold of negative sectional curvature to find “hidden symmetries”: Eberlein proves that a closed Riemannian manifold of negative sectional curvatures admitting infinitely many isometries of its universal cover must be locally symmetric.
From the dynamical perspective, hyperbolic dynamical systems also display such a phenomenon: if the centralizer group of a hyperbolic dynamical system is too large often it is conjugate to an algebraic one. In this talk we will consider hidden symmetries of the hyperbolic dynamical system given by the geodesic flow of a manifold of negative sectional curvatures. We will introduce an appropriate notion of a centralizer for the geodesic flow on the universal cover, and prove that when it is not discrete the metric must be locally symmetric.
On the holomorphic realization of branched covering maps
William Thurston’s theory of postcritically finite branched self-covers of the 2-sphere is an important breakthrough in modern complex dynamics. It explores the relationship between the topological properties of a branched cover, its dynamics, and the existence of a holomorphic ‘realization’ for the said branched cover. In this talk, we introduce this theory, and lay out progress in the last two decades to extend it to branched self-covers of the plane.