**Title: **Statistics of periodic points and a positive proportion Livsic theorem

**Speaker:** Caleb Dilsavor – Ohio State University

**Abstract: **The connection between the Ruelle-Perron-Frobenius operator and the statistics of a Hölder observable g with respect to an equilibrium state has a rich history, tracing back to an exercise in Ruelle’s book. A somewhat lesser known, but related, statistical theorem studied first by Lalley, and later by Sharp using the RPF operator, states that the periods of g grow approximately linearly with respect to length, with square rootoscillations chosen according to a normal distribution whose variance is equal to the (dynamical) variance of g. This result is known for aperiodic shifts of finite type, but surprisingly it is still notknown in full generality for their Hölder suspensions. I will describe a tentative result that fills in this gap, along with joint work with James Marshall Reber which uses this result to deduce a strengthening of Livsic’s theorem not previously considered: if a positive-upper-density proportion of the periods of g are zero, then g is in fact a coboundary.

**Zoom link:** https://osu.zoom.us/j/91943812487?pwd=K1lhTU02UTdMelBFTzhDdXRNcm80QT09

**Meeting ID:** 919 4381 2487

**Password:** Mixing

**Link of recorded talk:** https://osu.zoom.us/rec/share/ZiOZu_LJaCIMt0oBPGmFrenNVehsf2ZxaM8Myw1DiBNJ9cyVzrdFZHaqTIOoP3vO.ap18_rehC7ecOOgQ