## Medvedev degrees of 2-dimensional subshifts of finite type. Ergodic Theory and Dynamical Systems

Citations: | 17 - 9 self |

### BibTeX

@MISC{Simpson_medvedevdegrees,

author = {Stephen G. Simpson},

title = {Medvedev degrees of 2-dimensional subshifts of finite type. Ergodic Theory and Dynamical Systems},

year = {}

}

### OpenURL

### Abstract

In this paper we apply some fundamental concepts and results from recursion theory in order to obtain an apparently new counterexample in symbolic dynamics. Two sets X and Y are said to be Medvedev equivalent if there exist partial recursive functionals from X into Y and vice versa. The Medvedev degree of X is the equivalence class of X under Medvedev equivalence. There is an extensive recursion-theoretic literature on the lattice of Medvedev degrees of nonempty Π 0 1 subsets of {0, 1} N. This lattice is known as Ps. We prove that Ps consists precisely of the Medvedev degrees of 2-dimensional subshifts of finite type. We use this result to obtain an infinite collection of 2-dimensional subshifts of finite type which are, in a certain sense, mutually incompatible. Definition 1. Let A be a finite set of symbols. The full 2-dimensional shift on A is the dynamical system consisting of the natural action of Z2 on the compact set AZ2. A 2-dimensional subshift is a nonempty closed set X ⊆ AZ2 which is invariant under the action of Z2. A 2-dimensional subshift X is said to be of finite type if it is defined by a finite set of forbidden configurations. An interesting paper on 2-dimensional subshifts of finite type is Mozes [22]. A standard reference for the 1-dimensional case is the book of Lind/Marcus [20], which also includes an appendix [20, §13.10] on the 2-dimensional case.