Factor maps between tiling dynamical systems

We show that there is no Curtis-Hedlund-Lyndon Theorem for factor maps between tiling dynamical systems: there are codes between such systems which cannot be achieved by working within a finite window. By considering 1-dimensional tiling systems, which are the same as flows under functions on subshifts with finite alphabets of symbols, we construct a `simple' code which is not `local', a local code which is not simple, and a continuous code which is neither local nor simple.Comment: 8 page

Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem

We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.Comment: Published at http://dx.doi.org/10.1214/074921706000000266 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Symmetric Gibbs measures

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures---a version of de Finetti's Theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field

Dynamical properties of the Pascal adic transformation

We study the dynamics of a transformation that acts on infinite paths in the graph associated with Pascal's triangle. For each ergodic invariant measure the asymptotic law of the return time to cylinders is given by a step function. We construct a representation of the system by a subshift on a two-symbol alphabet and then prove that the complexity function of this subshift is asymptotic to a cubic, the frequencies of occurrence of blocks behave in a regular manner, and the subshift is topologically weak mixing