Factoring Higher-Dimensional Shifts Of Finite Type Onto The Full Shift

A one-dimensional shift of finite type with entropy at least log factors onto the full -shift. The factor map is constructed by exploiting the fact that , or a subshift of , is conjugate to a shift of finite type in which every symbol can be followed by at least symbols. We will investigate analogous statements for higher-dimensional shifts of finite type. We will also show that for a certain class of mixing higher-dimensional shifts of finite type, sufficient entropy implies that is finitely equivalent to a shift of finite type that maps onto the full -shift

Directional Recurrence For Infinite Measure Preserving Zᵈ Actions

We define directional recurrence for infinite measure preserving Zd actions both intrinsically and via the unit suspension flow and prove that the two definitions are equivalent. We study the structure of the set of recurrent directions and show it is always a Gδ set. We construct an example of a recurrent action with no recurrent directions, answering a question posed in a 2007 paper of Daniel J. Rudolph. We also show by example that it is possible for a recurrent action to not be recurrent in an irrational direction even if all its sub-actions are recurrent

Finite Rank Zᵈ Actions And The Loosely Bernoulli Property

We define finite rank for Zᵈ actions and show that those finite rank actions with a certain tower shape are loosely Bernoulli for d ≥ 1

Auslander Systems

The authors generalize the dynamical system constructed by J. Auslander in 1959, resulting in perhaps the simplest family of examples of minimal but not strictly ergodic systems. A characterization of unique ergodicity and mean-L-stability is given. The new systems are also shown to have zero topological entropy and fail to be weakly rigid. Some results on the set of idempotents in the enveloping semigroup are also achieved

Even Kakutani Equivalence And The Loose Block Independence Property For Positive Entropy Zᵈ Actions

In this paper we define the loose block independence property for positive entropy Zᵈ actions and extend some of the classical results to higher dimensions. In particular, we prove that two loose block independent actions are even Kakutani equivalent if and only if they have the same entropy. We also prove that for d \u3e 1 the ergodic, isometric extensions of the positive entropy loose block independent Zᵈ actions are also loose block independent

The Symbolic Dynamics Of Multidimensional Tiling Systems

We prove a multidimensional version of the theorem that every shift of finite type has a power that can be realized as the same power of a tiling system. We also show that the set of entropies of tiling systems equals the set of entropies of shifts of finite type

Speedups And Orbit Equivalence Of Finite Extensions Of Ergodic Zᵈ-Actions

We classify n-point extensions of ergodic Zᵈ-actions up to relative orbit equivalence and establish criteria under which one n-point extension of an ergodic Zᵈ-action can be sped up to be relatively isomorphic to an n-point extension of another ergodic Zᵈ-action. Both results are characterized in terms of an algebraic object associated to each n-point extension which is a conjugacy class of subgroups of the symmetric group on n elements

Rank One And Loosely Bernoulli Actions In Zᵈ

We define rank one for Zᵈ actions and show that those rank one actions with a certain tower shape are loosely Bernoulli for d greater than or equal to 1. We also construct a zero entropy Z² loosely Bernoulli action with a zero entropy, ergodic, non-loosely Bernoulli one-dimensional subaction

IP Cluster Points Idempotents, And Recurrent Sequences

We consider the dynamical system (M,S) where M is the orbit closure of a nonperiodic recurrent sequence of O\u27s and 1\u27s (for example, the Morse sequence) and S is the shift map. The enveloping semigroup is E(M) = {Sⁿ : n ∈ Z} where the closure is taken in the topology of pointwise convergence. H. Furstenberg was the first to establish the existence of relationships between recurrence, IP sets, and idempotents in the enveloping semigroup, and the first author has proven that the closure of the set of idempotents coincides with the IP cluster points. In this paper the authors compute this set for (M,S) and shed light on other combinatorial properties of generalized Morse sequences