Krieger's generator theorem shows that any free invertible ergodic measure preserving action (Y,\mu, S) can be modeled by A^Z (equipped with the shift action) provided the natural entropy constraint is satisfied; we call such systems (A^Z here) universal. Along with Tom Meyerovitch, we establish general specification like conditions under which Z^d- shift spaces are universal. These conditions are satisfied by a wide variety of shift spaces. Using these ideas we recover a strong form of the Alpern's lemma for Z^d actions and prove that the space of proper colourings of the Z^d lattice and the domino tilings of the Z^2 lattice are universal (answering a question by Şahin and Robinson).

Discrete time fractal Brownian motion with Hurst parameter H> 3/4 has a local limit theorem conditioned on the future. This permits to apply Aaronson's extension of the Darling-Kac theory to derive convergence of local times of fBM towards a Mittag-Leffler distribution. The method of prove has a general formulation for stochastic processes once a local limit theorem at "0" is established. These results are jointly with Xiaofei Zheng.

In this lecture a recent joint work with Toshihiro Hamachi will be discussed, in which we prove that two noncommutative Bernoulli schemes with the same entropy are isomorphic. The underlying ideas of our proof are largely based on the finitary isomorphism theory developed with Meir Smorodinsky at this university in the 1970's. These ideas can best be understood by recalling the important first example of Meshalkin (1964), in which he showed that the commutative Bernoulli schemes based on the probability vectors (1/4,1/4,1/4,1/4) and (1/2,1/8,1/8,1/8,1/8) are isomorphic. We also obtain a factor theorem for unequal entropies.

The liftable centralizer for special flows over irrational rotations is studied. It is shownthat there are such flows under piecewise constant roof functions which are rigid and whose liftable centralizer is trivial.The talk is based on a joint work with J.-P. Conze

TBA.

The notion of the critical iterate was first introduced by Cruz and da Rocha (2005). We will see that for any given singularity order vector and any marked singularity, there exists a piecewise rotation of the circle such that one of discontinuous points and one of its associated critical iterate generate a translation surface which has the given singularity orders with the given marked singularity. (joint work with Kae Inoue)

A potential function over a Markov shift with infinite number of states can be classified as either recurrent or transient. It is known that in the recurrent case, the eigenmeasure and the eigenfunction of the Ruelle operator exist and are unique and the eigenmeasure is conservative. However, if the potential is transient, then there exist eigenmeasures and eigenfunctions which are not necessarily unique and the eigenmeasures are totally dissipative. In this talk, we will show that these eigenmeasures and eigenfunctions can be fully characterized by a suitable Martin boundary and present a duality between the two.

TBA.