We will recall some basic ergodic and statistical properties such as: ergodicity, (quantitative) mixing, K property, Bernoullicity, central limit theorem. We will be interested in rigidity and flexibility of these properties for smooth diffeomorphisms preserving a smooth measure. Our main rigidity result is that C^{1+\alpha} smooth diffeomorphisms which are exponentially mixing are Bernoulli (joint with D. Dolgopyat and F.Rodriguez-Hertz). For flexibility results we show existence of C^r smooth systems which satisfy the (non-trivial) central limit theorem and are of zero entropy. Moreover we show that there are smooth K, non-Bernoulli systems which satisfy (non-trivial) central limit theorem (joint with D. Dolgopyat, C. Dong, P.Nandori).

Zoom link: https://iu.zoom.us/j/661711533?pwd=RTFVTjMrQ1pYTCtIZzIvVGVvODV2QT09
password is 076877 if needed. Speaker(s): Adam Kanigowski (University of Maryland)