# An example of a deterministic cellular automaton exhibiting linear-exponential convergence to the steady state

## TL;DR

This paper rigorously analyzes a 3-state cellular automaton that demonstrates linear-exponential convergence to a steady state, expanding understanding of degenerate hyperbolic behavior in cellular automata.

## Contribution

The paper provides a rigorous derivation of density formulas for the automaton, moving beyond previous semi-empirical approaches and analyzing cluster interactions.

## Key findings

- Confirmed linear-exponential convergence to steady state.
- Derived exact density formulas for states 0, 1, and 2.
- Enhanced understanding of degenerate hyperbolic behavior in cellular automata.

## Abstract

In a recent paper [arXiv:1506.06649 [nlin.CG]], we presented an example of a 3-state cellular automaton which exhibits behaviour analogous to degenerate hyperbolicity often observed in finite-dimensional dynamical systems. We also calculated densities of 0, 1 and 2 after n iterations of this rule, using finite state machines to conjecture patterns present in preimage sets. Here, we re-derive the same formulae in a rigorous way, without resorting to any semi-empirical methods. This is done by analysing the behaviour of continuous clusters of symbols and by considering their interactions.