Given a topologically mixing countable-state Markov shift Σ, Sarig showed that every weakly Hölder continuous potential φ:Σ→ ℝ with finite Gurevich pressure, which satisfies the strong positive recurrence property, has a unique Ruelle–Perron–Frobenius measure 𝜇φ. This measure can be interpreted as the unique equilibrium state for φ. For subshifts of finite type, Coelho and Quas established that the map φ↦𝜇φ is continuous with respect to the Hölder metric on the space of potentials and the d̅ -metric on the space of measures. This metric preserves useful ergodic properties. In this talk, we will discuss the extension of this result to countable-state Markov shifts.
In this talk, we will discuss the average value taken by a knot invariant on the periodic orbits of a hyperbolic flow on the 3-sphere.
The first relevant result comes from the work of Contreras, who studied the average linking number between periodic orbits. Contreras found precise asymptotic growth rates for this number, as the period tends to infinity. In the proof, the Gauss linking integral is used to translate the problem into the language of ergodic theory.
In recent work, we instead consider the average value of a Vassiliev invariant on periodic orbits. Here, the configuration space integrals of Bott and Taubes take the place of the Gauss linking integral in Contreras' work.
A surface S of negative Euler characteristic admits many different hyperbolic structures. These are organized by the classical Teichmüller space of S. Each hyperbolic structure determines (and is determined by) a length spectrum function, which associates to each homotopy class of closed curves on S, the length of the geodesic representative. One way to tell a pair of hyperbolic structures apart is to compare their length spectra; this idea leads to Thurston’s asymmetric metric on Teichmüller space. We will recall some key points of Thurston’s story and then generalize to find interesting geometric/dynamical invariants associated to a d-tuple of hyperbolic structures (d > 2) and, most generally, to a positive representation of π1 S into a real split, semi-simple Lie group of higher rank. Joint work with François Guéritaud and Fanny Kassel.
Jeff's first talk is the colloquim talk at UIC. You can find his abstract and title at this link.
In this talk, we will examine the correlation of length spectra between two geometric structures on (potentially cusped) surfaces. This correlation aims to capture both the distinctions and commonalities between these structures. Our geometric findings stem from their symbolic counterparts. Furthermore, we will delve into the geometric interpretation of the correlation number and its relationship with the Manhattan curve.
The geometric structures of our interest are cusped Hitchin representations, including cusped Fuchsian representations (a.k.a. cusped hyperbolic metrics). This is joint work with Giuseppe Martone.
In this talk I’ll begin by telling a little bit of Thurston’s beautiful story connecting the dynamics of finite-type surface homeomorphisms with the geometry of 3-manifolds. I will then share some more recent work of myself and others which connect the dynamics of infinite-type surface homeomorphisms with the geometry of certain 3-manifolds.
In this talk, given a (hyperbolizable) surface S (with boundary) we will describe the action of the mapping class group Mod(S) on the (relative) SL(2, C)–character variety X(S) := Hom(π1(S), SL(2, C))/SL(2, C), and define an open domain of discontinuity of the action which (in many cases) strictly contains the interior of the set of discrete and faithful representations.
In the first part of the talk, we will focus on the original idea of Bowditch and consider S=S_{1,1} the once-holed torus. We will describe the combinatorial methods suggested by Bowditch based on the relationship between points in the SL(2, C)–character variety X(S) and Markoff triples, the main steps of the proofs and its generalization by Tan-Wong-Zhang.
In the second part of the talk, we will focus on joint work with Palesi, Tan, and Lawton, and generalize the description to various cases, considering S = S_{0,4} the four-holed sphere, S= N_{1,3} the three-holed projective plane and replacing SL(2,C) with SU(2,1). For each of these generalizations, we will explain how to modify the set-up discussed in the first part of the talk, and the difficulties that arise when one tries to do a general discussion. Time-permitting, we will also discuss relations with the work of Minsky on primitive-stable representations and Schlich.
Sara's first talk is a Big Ideas talk.
How can divisible convex sets exhibit negative curvature properties without being negatively curved themselves? In this talk we will discuss work-in-progress concerning the β-uniform convexity of a Finsler metric on strictly convex divisible domains.
The study of the geometries of surfaces is classical, and so is the study of homeomorphisms of surfaces and their dynamics. These aspects of low dimensional topology, geometry and dynamics came together starting with the work of Thurston and revolutionised low dimensional topology. Recently, there has been a surge in interest in studying infinite type surfaces. We will see how to try and adapt the known theories of geometry and dynamics to infinite type surfaces.
Sharkovsky’s Theorem is a classical result on the forcing of periodic orbits of interval maps. In this talk, we will consider n-Sharkovsky orderings which describe orbit forcing of maps on a tree. In this talk, I will explain how the Sharkovsky ordering and n-Sharkovsky orderings are found in the Mandelbrot set, along principal and non principal veins. Moreover I will discuss the surprisingly straightforward dynamics of some coordinating parameters along these veins.