Non-expansive directions for $Z^2$-actions

TL;DR

This paper demonstrates that every direction in the plane can be realized as a unique non-expansive direction in a -action, and explores cellular automata with specific sensitivity and Lyapunov exponent properties.

Contribution

It answers Boyle and Lind's question by constructing -actions with prescribed non-expansive directions and analyzes cellular automata with zero Lyapunov exponents exhibiting sensitivity.

Findings

Every plane direction is a unique non-expansive direction for some -action.

Constructed cellular automata can have zero Lyapunov exponents while remaining sensitive.

For any positive , there exists a cellular automaton with symmetric Lyapunov exponents .

Abstract

We show that any direction in the plane occurs as the unique non-expansive direction of a \mathbb{Z}^{2} action, answering a question of Boyle and Lind. In the case of rational directions, the subaction obtained is non-trivial. We also establish that a cellular automaton can have zero Lyapunov exponents and at the same time act sensitively; and more generally, for any positive real \theta there is a cellular automaton acting on an appropriate subshift with \lambda^{+}=-\lambda^{-}=\theta.