Series: Penn State Logic Seminar

Date: Tuesday, November 4, 2003

Time: 2:30 - 3:45 PM

Place: 324 Sackett Building (note unusual location)

Speaker: Ksenija Simic, Carnegie Mellon University, Mathematics


The Mean Ergodic Theorem in Weak Subsystems of Second Order Arithmetic


The mean ergodic theorem states that for an appropriately defined
measure preserving transformation T on a space X, the sequence
S_n=(1/n)sum_{k=0}^{n-1}f(T^k) converges in the 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.