** Joinings of dynamical systems **

We deal with probability measure preserving maps.

Measures control points. For instance, Birkhoff’s theorem states that time averages of a function converge a.e. In the uniquely ergodic case, convergence holds everywher for continuous functions.

**1. Joinings **

A *joining* is an invariant measure on the product of two dynamical systems, whose marginals are the given measures.

The product measure is always a joining. If it is the only one, one says that the dynamical systems are *disjoint*. In this case, and if both systems are uniquely ergodic, the product system is uniquely ergodic.

If two systems are isomorphic, the graph of the conjugating map defines a joining.

Let us call *graph joinings* the joinings of system to itself provided by graphs of maps commuting with the dynamics.

**2. Interval exchange transformations **

This is a generalization of circle rotations. Cut an interval into finitely many intervals and permute them. In general, there is a unique ergodic measure.

The geodesic flow of a translation surface, for each fixed direction, determines an interval exchange transformation. This arises in polygonal billiards with rational angles. By changing the speed of the flow, one can smooth it and get a smooth flow on a surface. Conversely, every smooth flow on a surface is a time change of a geodesic flow of a translation surface.

Theorem 1 (Masur-Veech)For a.e. translation surface, the vertical flow is uniquely ergodic.

Theorem 2 (Chaika)A.e. pair of interval exchange transformations are disjoint.

Theorem 3 (Chaika-)A.e. pair of translation surfaces are disjoint.

It follows that products are almost always uniquely ergodic. A.e. point is recurrent, and the first return map is a rectangle exchange map.

**Question (Gowers)**. Does every rectangle exchange have only recurrent points ?

I think counterexamples should be found among products of an IET with itself.

**3. Do circles equidistribute? **

Yes on the torus. Indeed, large circles track lines with irrational slopes. This fails for translation surfaces, since arcs break into typically very small arcs.

Theorem 4 (Chaika-Hubert)For a.e. translation surface, for all points , there is a set of density 1 such that averages on circles converge as radius tends to infinity.

This follows from unique ergodicity of products. Indeed, pick directions. Due to law of large numbers, they reflect the behaviour of all directions. Unique ergodicity allows to pick a special starting point like .