Georgia Institute of Technology College of Sciences

Consider two volume-preserving, smooth diffeomorphisms f and g of a compact manifold M.  Define the random walk on M by selecting either f or g (i.i.d.) at each iterate.  A number of questions arise in this setting:

1. What are the closed subsets of M invariant under both f and g?
2. What are the stationary measures on M for the random walk.  In particular, are the stationary measures invariant under f and g?

Conjecturally, for a generic pair of f and g we should be able to answer the above.  I will describe one sufficient criteria on f and g underwhich we can give some partial answers to the above questions.  Such a criteria is expected to be generic amoung pairs of (volume-preserving) diffeomorphisms and should be able to be verified in a number of naturally occurring geometric settings where the above questions are not fully answered.