The Pugh-Shub conjectures have been a major motivation for recent research into partially hyperbolic dynamics. Roughly speaking, these conjectures state that most volume preserving partially hyperbolic systems are stably ergodic and that accessibility is the key tool for establishing this. The Pugh-Shub conjectures have been proven for systems in dimension 3, and this leads to the question of exactly which systems in dimension 3 are accessible. In my talk, I will discuss progress on answering this question.

In this talk I will share some pleasant memories I have had about Charles Pugh in more than 30 years, and how much my colleagues and I are impressed by his remarkable mathematical work as well as his nice personality. I also include a little piece of story about the C¹ closing lemma.

We'll briefly explore the possible existence of non-ergodic C¹ Anosov diffeomorphisms, inspired by a non-exapmple provided by Bowen ('75) in the form of a horseshoe with positive measure on T².

A hundred years ago, Besicovitch constructed sets in the plane which contain a unit line segment in every direction but have arbitrarily small area, or even with zero area. Similarly, one can construct a set in R^n with a unit line segment in every direction and volume zero. The Hausdorff dimension of Besicovitch's construction is n, the maximum possible value. In the plane, the Hausdorff dimension of such a set is always full. In higher dimensions, the problem is wide open. It attracted a lot of attention because of connections to problems in Fourier analysis like the restriction problem. We will try to say something about what makes the problem difficult.

For Axiom A diffeomorphisms we know there exist finitely many ergodic measures of maximal entropy and these are supported on basic sets. In the partially hyperbolic setting which behavior wins - the center behavior or hyperbolic behavior? Although there are partial answers, the overall picture is still unclear. We will explain some of the examples and difficulties.

This question is one part of the puzzle connecting the topology and geometry of a manifold to the possible dynamical systems that it supports. I like to think about the topologists' version of this, considering flows up to orbit equivalence. Some variations, replacing "hyperbolic" with another geometric or topological condition have a known answer like "zero" or "at most two up to orbit equivalence" or "typically infinitely many" but the hyperbolic case is fascinating and still pretty much the wild west.

How do you draw a picture of the wind? I'll talk about some work in data visualization, focusing on the problems that arise—mathematical and non-mathematical—in trying to create a living portrait of air currents across the country. It turns out that a smooth two-dimensional flow is surprisingly hard to portray faithfully.

I will discuss the problem of the regularity of curves that are invariant by a symplectic twist map.

The set of periodic points of a diffeomorphism can be used as skeleton to analyzing the dynamics. It is a major issue to know its size. For that reason, Charles Pugh’s closing lemma has been fundamental for our understanding of the C¹- dynamics.

The next step is to describe how the periodic orbits are organized: I will illustrate this with some examples and questions.

Basic ideas in monotone and chaotic dynamics are reviewed, and recent theorems stated. Lorenz's system of differential equations having a chaotic attractor and Devaney's axioms for chaos are reviewed. A new theorem is stated, showing that a broad class of monotone maps do not have chaotic attractors. It is conjectured that this holds for all monotone maps.