Randomness and ergodic theorems under measure-preserving transformations
Ergodic theorems describe regular measure-theoretic behavior, and a point is said to be algorithmically random if it has no rare measure-theoretic properties of a certain kind. The connections between these two types of regularity have been well studied over the past five years. I will discuss Birkhoff’s ergodic theorem with respect to transformations that are measure-preserving but not necessarily ergodic in the context of a computable probability space. Then I will show that each point in such a space that is not Martin-L”of random does not satisfy Birkhoff’s ergodic theorem with respect to every computable set and measure-preserving transformation.
This work is joint with Henry Towsner.
Prof. Franklin has been an Assistant Professor in the mathematics department of Hofstra University since 2014. She studies algorithmic randomness and recursion theory, with applications in probability and ergodic theory. She received her doctorate from the University of California-Berkeley, under the supervision of Ted Slaman, and has taught at the University of Connecticut, Dartmouth College, the University of Waterloo, and the National University of Singapore.