Minicourses

We will first introduce the main concepts and questions of thermodynamic formalism in the setting of one-dimensional holomorphic dynamics. We will see how the complex setting often allows us to overcome the need for hyperbolicity assumptions usually made to study these problems in more general settings. We will then move to higher dimensions. Holomorphic dynamical systems in higher dimensions exhibit different behaviors from those in dimension 1, as holomorphic maps are not necessarily conformal anymore and several classical theorems in complex analysis no longer hold in several complex variables. We will introduce a volume dimension for measures with positive Lyapunov exponents as a dynamical replacement for the Hausdorff dimension and discuss its applications.

Slides 1, Slides 2, Video 1, Video 2, Video 3, Video 4 

Lecture notes (typed up by Alexander Paschal)

We consider the (semi)group action of SL3(R) on P(R3), a primary example of non-conformal, non-linear, and non-strictly contracting action. The goal of these lectures is to present recent progress for studying the Hausdorff dimension of a dynamically defined limit set in P(R3), focusing on the result of generalizing the classical Patterson-Sullivan formula using the approach of stationary measures.

Lecture 1 Introduction and statement of the main results (Pan) Video 1 

Lecture 2,3 Entropy growth argument for Bernoulli convolution based on [Hoc14]; non-concentration of stationary measures on arithmetic sequence via Fourier decay (Li) Video 2,  Video 3

Lecture 4 Variational principle for Anosov representations; proof of the dimension formula of limit sets [LPX23] [JLPX23] (Pan) Video 4

[Hoc14] Michael Hochman. On self-similar sets with overlaps and inverse theorems for entropy. Annals of Mathematics, 180(2):773–822, September 2014.

[JLPX23] Yuxiang Jiao, Jialun Li, Wenyu Pan, and Disheng Xu. On the dimension of limit sets on P(R3) via stationary measures: Variational principles and applications. to appear in IMRN, (arXiv:2311.10262), December 2023.

[LL23] François Ledrappier and Pablo Lessa. Exact dimension of Furstenberg measures. Geometric and Functional Analysis, 33(1):245–298, February 2023.

[LPX23] Jialun Li, Wenyu Pan, and Disheng Xu. On the dimension of limit sets on P(R3) via stationary measures: The theory and applications. (arXiv:2311.10265), 2023.

[Rap21] Ariel Rapaport. Exact dimensionality and Ledrappier-Young formula for the Furstenberg measure. Transactions of the American Mathematical Society, 374(7):5225–5268, April 2021.

Lecture notes (typed up by Ana de Araujo, Jamerson Bezerra, and Graccyela Salcedo)

Lecture notes (typed up by Yuxiang Jiao)

The talks have ideas from works with D. Damjanovic, R. Spatzier, A. Uzman and D. Xu in various combinations.

Talk 1: Fundamentals of (partially) hyperbolic abelian group actions

We'll revisit some fundamental ideas in (partially) hyperbolic dynamics for abelian group actions, and in some cases, their limitations in higher-rank. In particular, we will consider strengths and defects of the Anosov closing lemma, transitivity assumptions, uniform vs non-uniform coarse Lyapunov foliations, canonical invariant measures.  Video 1

Talk 2: Tools for rigidity

In this talk, we'll describe some tools for building homogeneous structures from hyperbolic actions. In particular, we'll see how the structures from the first talk interface with other developments in partially hyperbolic dynamics, including accessibility, the invariance principle, and Brin-Pesin bundle extensions, to get a rigidity theorem for abelian group actions. Video 2

Talk 3: Non-rigidity and partial rigidity

In this talk, we'll investigate some situations in which the expected rigidity phenomena fail. This will involve time changes which highlight the nuances of the fundamentals, and a complete description of a certain class of hyperbolic abelian group actions, even in the presence of rank one factors, where rigid and non-rigid phenomena both emerge. Video 3

The course will focus on rigidity of u-Gibbs measures and the tools needed to establish such results.  The course will be organized roughly around the following topics: (1) u-Gibbs measures: Random dynamics, stationary measures, skew extensions, u-invariance; SL(2,R)-actions and P-invariant measures; uu-gibbs for flows and maps admitting dominated splitting (2) Homogeneous structures: normal forms and their construction; disintegration of measure along measurable partitions and locally finite measures along unstable manifolds; homogeneity of measures along unstable manifolds.  (3) Extra invariance of measures: standard arguments to obtain extra invariance, modifications outside the setting of conformal dynamics.  (4) ideas from proof of Brown-Eskin-Filip-Rodriguez Hertz: space of generalized u-Gibbs measures and induced cocycle, quantitative transversality, entropy estimates.

One defines unstable entropies by focussing on the dynamics inside the leaves of unstable foliations. We will focus on the uniformly expanding case and especially the strongly unstable foliations of partially hyperbolic diffeomorphisms. As we will explain, this focus yields many strong properties generalizing the much simpler dynamics of expanding maps, such as the relation between thermodynamic formalism and geometry of measures. Finally, we will show how unstable entropy is key to a selection of recent results about both measures maximizing the entropy and Sinai-Ruelle-Bowen measures for partially hyperbolic systems. 

Our plan for the lectures is as follows:

• definition of measured and topological entropies for expanding foliations; variational principles;

• identification of conditional measures and u-states; the invariance principle;

• applications to partial hyperbolic dynamics 

• • SRB measures for mostly contracting/expanding diffeomorphisms;

  • MME for perturbations of time-one maps of Anosov flows.

Slides talk 1, Slides talk 4

Video 1, Video 2, Video 3, Video 4  

This mini-course discusses methods for measuring the complexity of classifying dynamical systems.

Classification problems examine when two dynamical systems are equivalent (conjugate or isomorphic). To solve such problems, we must characterize all possible equivalence classes of the isomorphism relation. We will explore tools for proving when a classification problem is unsolvable—when it becomes impossible to classify certain systems under specific equivalence relations using countable techniques.

We will employ descriptive set theory to formalize concepts like "solvability of classification problems" or "countable techniques. " A central concept is a Borel reduction, which allows us to establish that one classification problem is at most as complex as another.

Using Borel reductions, we organize classification problems into a hierarchy based on their inherent complexity.

No prior knowledge of descriptive set theory is required for this course.

Lecture 1: The Toolbox

We begin with an impossible goal: surveying the rapidly expanding theory of complexity of equivalence relations. This theory forms the foundation for (anti-)classification results in dynamics.

Lecture 2: The Scoreboard

We will discuss various classification and anti-classification results in dynamical systems and ergodic theory.

Lecture 3: The Action

We will analyze specific proofs of anti-classification results.

Core References:

Jérôme Buzzi et al. "Open questions in descriptive set theory and dynamical systems." arXiv:2305.00248 [math.DS] (2023).

Konrad Deka et al. "Bowen's Problem 32 and the conjugacy problem for systems with specification."

Matthew Foreman, "What is a Borel reduction?" Notices Amer. Math. Soc. 65 (2018), no. 10, 1263–1268.

Matthew Foreman, "The complexity and the structure and classification of dynamical systems." Ergodic theory, 529–576, Encycl. Complex. Syst. Sci., Springer, New York, (2023).

Greg Hjorth, "Borel Equivalence Relations." In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht (2010).

Video 1, Video 2 , Video 3