Statistical Mechanics of Surjective Cellular Automata

Reversible cellular automata are seen as microscopic physical models, and their states of macroscopic equilibrium are described using invariant probability measures. We establish a connection between the invariance of Gibbs measures and the conservation of additive quantities in surjective cellular automata. Namely, we show that the simplex of shift-invariant Gibbs measures associated to a Hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the Hamiltonian. A special case is the (well-known) invariance of the uniform Bernoulli measure under surjective cellular automata, which corresponds to the conservation of the trivial Hamiltonian. As an application, we obtain results indicating the lack of (non-trivial) Gibbs or Markov invariant measures for "sufficiently chaotic" cellular automata. We discuss the relevance of the randomization property of algebraic cellular automata to the problem of approach to macroscopic equilibrium, and pose several open questions. As an aside, a shift-invariant pre-image of a Gibbs measure under a pre-injective factor map between shifts of finite type turns out to be always a Gibbs measure. We provide a sufficient condition under which the image of a Gibbs measure under a pre-injective factor map is not a Gibbs measure. We point out a potential application of pre-injective factor maps as a tool in the study of phase transitions in statistical mechanical models.Comment: 50 pages, 7 figure

Defect Particle Kinematics in One-Dimensional Cellular Automata

Let A^Z be the Cantor space of bi-infinite sequences in a finite alphabet A, and let sigma be the shift map on A^Z. A `cellular automaton' is a continuous, sigma-commuting self-map Phi of A^Z, and a `Phi-invariant subshift' is a closed, (Phi,sigma)-invariant subset X of A^Z. Suppose x is a sequence in A^Z which is X-admissible everywhere except for some small region we call a `defect'. It has been empirically observed that such defects persist under iteration of Phi, and often propagate like `particles'. We characterize the motion of these particles, and show that it falls into several regimes, ranging from simple deterministic motion, to generalized random walks, to complex motion emulating Turing machines or pushdown automata. One consequence is that some questions about defect behaviour are formally undecidable.Comment: 37 pages, 9 figures, 3 table

Density of periodic points, invariant measures and almost equicontinuous points of cellular automata

Revisiting the notion of m-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure m by iterations of a m-almost equicontinuous automata F, converges in Cesaro mean to an invariant measure mc. If the initial measure m is a Bernouilli measure, we prove that the Cesaro mean limit measure mc is shift mixing. Therefore we also show that for any shift ergodic and F-invariant measure m, the existence of m-almost equicontinuous points implies that the set of periodic points is dense in the topological support S(m) of the invariant measure m. Finally we give a non trivial example of a couple (m-equicontinuous cellular automata F, shift ergodic and F-invariant measure m) which has no equicontinuous point in S(m)