We present two examples of finite-alphabet, infinite excess entropy processes generated by invariant hidden Markov models (HMMs) with countable state sets. The first, simpler example is not ergodic, but the second is. It appears these are the first constructions of processes of this type. Previous examples of infinite excess entropy processes over finite alphabets admit only invariant HMM prese...

Given a TQFT in dimension d + 1, and an infinite cyclic covering of a closed (d + 1)-dimensional manifold M , we define an invariant taking values in a strong shift equivalence class of matrices. The notion of strong shift equivalence originated in R. Williams’ work in symbolic dynamics. The Turaev-Viro module associated to a TQFT and an infinite cyclic covering is then given by the Jordan form...

Annals of Probability 31(2003), 2109-2135. Invariant man-ifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invarian...

Abstract We consider general infinite-dimensional dynamical systems with the Galilean and spatiotemporal scaling symmetry groups. Introducing equivalence relation respect to temporal scalings transformations, we define a representative set containing single element within each class. Temporal transformations do not commute evolution operator (flow) and, hence, is invariant. Despite of that, pro...

We provide an N/V -limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on R, d ≥ 1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ ⊂ R with finite volume (Lebesgue measure) V = |Λ| < ∞. The aim is to approximate the infinite particle, infinit...