Is the injectivity of the global function of a cellular automaton in the hyperbolic plane undecidable?

TL;DR

This paper investigates the undecidability of the injectivity of the global function of cellular automata in the hyperbolic plane, extending known results from Euclidean cases under certain configuration restrictions.

Contribution

It provides a partial answer showing undecidability of injectivity in the hyperbolic plane for restricted configurations, building on prior Euclidean results.

Findings

Undecidability of injectivity in hyperbolic plane under certain conditions

Extension of Euclidean plane results to hyperbolic geometry

Partial undecidability result for restricted configuration classes

Abstract

In this paper, we look at the following question. We consider cellular automata in the hyperbolic plane and we consider the global function defined on all possible configurations. Is the injectivity of this function undecidable? The problem was answered positively in the case of the Euclidean plane by Jarkko Kari, in 1994. In the present paper, we give a partial answer: when the configurations are restricted to a certain condition, the problem is undecidable.