Aimee S. A. Johnson

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.

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

A one-dimensional shift of finite type (X,Z) with entropy at least logn factors onto the full n-shift. The factor map is constructed by exploiting the fact that X, or a subshift of X, is conjugate to a shift of finite type in which every symbol can be followed by at least n symbols. We will investigate analogous statements for higherdimensional 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 (X,Z ) is finitely equivalent to a shift of finite type that maps onto the full n-shift.

Putting The Pieces Together: Understanding Robinson’s Nonperiodic Tilings

Aimee Johnson (aimee@swarthmore.edu) obtained her B.A. from the University of California, Berkeley, in 1984 and her Ph.D. from the University of Maryland, College Park, in 1990. She is now part of the Department of Mathematics and Statistics at Swarthmore College. Her research interests are ergodic theory and symbolic dynamics. It is in the latter context that she and her coauthor came across the undecidability question for tilings that motivates this paper.

Commuting endomorphisms of the circle

In this paper the results of Shub and Sacksteder are extended to the following theorem: let ƒ 1 and ƒ 2 be two commuting, expansive, orientation-preserving maps of the circle with a common fixed point and with both in C 1+e or C r , r ≥2. Assume ƒ 1 is p -to-1 and ƒ 2 is q -to-1 where p and q generate a nonlacunary semigroup. Then there exists a diffeomorphism g of the same class such that g ƒ 1 g −1 = T P and g ƒ 2 g −1 = T q .

Renewal systems, sharp-eyed snakes, and shifts of finite type

1. INTRODUCTION. A common theme in all of mathematics is deciding when two seemingly different objects are actually the same in some sense. For example, every elementary school student knows that, despite outward appearances, the fractions 2/3 and 10/15 are the same. The calculus student knows that the functions f (x) = 3x and g(x) = (3x

Directional recurrence for infinite measure preserving actions

We define directional recurrence for infinite measure preserving

Recurrent sequences and IP sets

\mathbb{Z}^{d}

Speedups of ergodic group extensions of -actions

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

Isometric Extensions of zero entropy ℤ^{} Loosely Bernoulli Transformations

G_{{\it\delta}}

The Decomposition Theorem For Two-Dimensional Shifts Of Finite Type

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.