In the first session, we will introduce interval exchange transformations and discuss some of their typical properties; a typical IET is uniquely ergodic and weakly mixing, it is never strongly mixing. We will postpone the proofs to the second talk. Recall that a dynamical system is weakly mixing if the Cesaro averages of correlations tend to zero. We will show that a dichotomy holds regarding the speed of decay of Cesaro averages of sufficiently smooth observables. The rate is polynomial for IETs that can be suspended to give flows on surfaces of genus at least two (non-rotation class IETs) and is logarithmic for rotation type IETs (the ones whose corresponding suspension surface is always a Torus). We also show a logarithmic lower bound for the decay of Cesaro averages of correlations for rotation-type IETs thereby ruling out the possibility of having a polynomial rate of decay in this case.


[IPM Youth Seminars on Topology and Dynamics]