# Complexity of Short Rectangles and Periodicity

## Publication Date

2016

## Description

The Morse-Hedlund Theorem states that a bi-infinite sequence eta in a finite alphabet is periodic if and only if there exists n is an element of N such that the block complexity function P-eta(n) satisfies P-eta(n) <= n. In dimension two, Nivat conjectured that if there exist n, k is an element of N such that the n x k rectangular complexity P-eta(n, k) satisfies P-eta(n, k) <= nk, then eta is periodic. Sander and Tijdeman showed that this holds for k <= 2. We generalize their result, showing that Nivat's Conjecture holds for k <= 3. The method involves translating the combinatorial problem to a question about the nonexpansive subspaces of a certain Z(2) dynamical system, and then analyzing the resulting system. (C) 2015 Elsevier Ltd. All rights reserved.

## Journal

European Journal of Combinatorics

## Volume

52

## Issue

Pt. A

## First Page

146

## Last Page

173

## Department

Mathematics

## Link to Published Version

## DOI

10.1016/j.ejc.2015.10.003

## Recommended Citation

Cyr, Van and Kra, Bryna. "Complexity of Short Rectangles and Periodicity." *European Journal of Combinatorics* (2016)
: 146-173.