Larger Than Life's Invariant Measures

Larger than Life (LtL) is a four-parameter family of two-dimensional cellular automata that generalizes John Horton Conway's celebrated Game of Life (Life) to large neighborhoods and general birth and survival thresholds. If T is an LtL rule, and A a random configuration, then Tt(A) denotes the state of the system at time t starting from A. Tt(A) may be thought of as a Markov process since the sites update independently from all preceding times except the current one. The Markov process is degenerate since the transitions are deterministic. Nevertheless, it has a compact state space, so there exists a measure μ that is invariant under the rule. Since the dynamics are translation invariant, μ can be chosen so. In this paper, we prove that there are upper bounds, sometimes sharp, on the density of such measures. We also prove that there are upper bounds on the densities of LtL's still life measures, which are fixed points for given rules. Calculating these bounds requires a large neighborhood combinatorial calculation, which is done only for certain cases. The remaining cases are left as open problems.