Regularity of Non-stationary Stable Foliations and Applications to Spectral Properties of Sturmian Hamiltonians
We discuss regularity results concerning non-stationary stable foliations of hyperbolic maps satisfying a common cone condition. We then describe how to use these results to obtain properties concerning the dimension of the spectra of Sturmian Hamiltonians. This work is joint with Seung uk Jang.

(At a special time 11:30am)
Rational self-maps of the complex projective plane whose topological entropy is log of a transcendental number
In real dynamics it is easy to find mappings whose topological entropy is any given real number. For example, a tent map with slope of absolute value equal to s has topological entropy equal to log(s). In holomorphic dynamics, the situation is more complicated, with all previously known examples having topological entropy equal to log of an algebraic number. I will explain why this is so and then present a family of “toric” rational self-mappings of the complex projective plane that includes mappings whose topological entropy is log of a transcendental number. This is joint work with Jeffrey Diller and it builds on previous work of Bell-Diller-Jonsson.

Unique equilibrium states for Viana maps with small potentials
We investigate the thermodynamic formalism for Viana maps—skew products obtained by coupling an expanding circle map with a slightly perturbed quadratic family on the fibers. By applying general techniques developed by Climenhaga and Thompson, we show that for every Hölder potential whose oscillation is below an explicit threshold, an equilibrium state not only exists but is unique. All of these conclusions persist under sufficiently small perturbations of the reference map.

Deformation rigidity in proximity of de la Llave examples
De la Llave’s examples are certain skew-product Anosov diffeomorphisms that display a number of very interesting properties. In particular, they form an isospectral family that does not belong to a fixed smooth conjugacy class — a property reminiscent of the “you cannot hear the shape of a drum” counterexamples to Mark Kac’s question. I will explain that a generic perturbation of a De la Llave example does not admit such isospectral deformations. I will try to make it an accessible talk; the main player will be the local dynamics of a hyperbolic fixed point. Joint work with Martin Leguil.

Non-uniform specification approach to equilibrium states for parabolic rational maps
We develop the non-uniform specification approach to equilibrium states in the setting of parabolic rational maps. That is, for analytic endomorphisms of the Riemann sphere with rationally indifferent periodic points and no critical points in their Julia set. In this parabolic case, we demonstrate the first application of the Climenhaga-Thompson machinery in the complex dynamics setting. This is an alternative to the celebrated results on uniqueness, due to Denker and Urbanski which are based on transfer operator techniques. We extend the class of potentials from Hölder to those with the Bowen property on ‘good orbits’ and present an improvement on the pressure gap condition.

Margulis-like measures on expanding foliations: construction and rigidity
Given a diffeomorphism preserving a one-dimensional expanding foliation 𝓕 with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with controlled Jacobian and a Gibbs property. We then prove that for any measure of maximal u-entropy, its conditional measures on each leaf must be equivalent to the reference measures. When the measure of maximal u-entropy is a Gibbs 𝓕-state (i.e., when the reference measures are equivalent to the leafwise Lebesgue measure), we prove that the log-determinant of f must be cohomologous to a constant. We will discuss several applications, including the strong and center foliations of Anosov diffeomorphisms, factor over Anosov diffeomorphisms, and perturbations of the time-one map of geodesic flows on surfaces with negative curvature. We will also discuss several conjectures on the unique ergodicity and (exponential) equidistribution for the strong unstable foliation of an Anosov system. Joint with J. Buzzi, Y. Shi, and J. Yang.