#### Title

Nonexpansive Z(2)-Subdynamics and Nivat's Conjecture

#### Publication Date

2015

#### Journal

Transactions of the American Mathematical Society

#### Volume

367

#### Issue

9

#### First Page

6487

#### Last Page

6537

#### Department

Mathematics

#### Abstract

For a finite A, and eta: Z -> A the Morse-Hedlund Theorem states that eta is periodic if and only if there exists n E N such that the block complexity function P-eta(n) satisfies P-eta(n) <= n, and this statement is naturally studied by analyzing the dynamics of a Z-action associated with ai. In dimension two, we analyze the subdynamics of a Z(2)-action associated with eta: Z(2) -> A and show that if there exist n,k is an element of N such that the ii x k rectangular complexity (n, k) satisfies P,7(n k) < nk, then the periodicity of eta is equivalent to a statement about the expansive subspaces of this action. As a corollary, we show that if there exist n, k E N such that P-eta(n, k) <= nk/2, then eta is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words.

#### Link to Published Version

http://www.ams.org/journals/tran/2015-367-09/S0002-9947-2015-06391-0/S0002-9947-2015-06391-0.pdf

#### Recommended Citation

Cyr, Van and Kra, Bryna. "Nonexpansive Z(2)-Subdynamics and Nivat's Conjecture." *Transactions of the American Mathematical Society* 367, no. 9 (2015)
: 6487-6537.