Ernest Schimmerling

### Mathematical logic seminar - November 11, 2003

 Time: 12 - 1:20 p.m. Room: PPB 300 Speaker: Ksenija Simic Graduate Student in Mathematical Sciences Carnegie Mellon University Title: The Mean Ergodic Theorem in Weak Subsystems of Second Order Arithmetic Abstract: The mean ergodic theorem states that for an appropriately defined measure preserving transformation $T$ on a space $X$, the sequence $$S_{n}=\frac{1}{n}\sum_{k=0}^{n-1}f(T^{k})$$ converges in $L_2$ norm for all $f\in L_2(X)$. Due to the restrictions second order arithmetic imposes, it is not possible to define $T$ pointwise. Instead, we define it as a norm preserving linear operator on $L_2(X)$. As it transpires, it is more convenient to state and prove the theorem for the more abstract case - that of Hilbert spaces, following the approach of Halmos. A number of results from Hilbert space theory then needs to be established, before proving the actual theorem. I will give a brief overview of some of these results, and focus on the proof of the mean ergodic theorem. Finally, I will show that the mean ergodic theorem is equivalent to arithmetic comprehension over the base theory $RCA_0$. Organizer's note: Please bring your lunch.