Conservation Laws and Invariant Measures in Surjective Cellular Automata

We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.

[1]  A. Robinson.  I. Introduction , 1991 .

[2]  Enrico Formenti,et al.  Number conserving cellular automata II: dynamics , 2003, Theor. Comput. Sci..

[3]  D. Lind.  Applications of ergodic theory and sofic systems to cellular automata , 1984 .

[4]  Enrico Formenti,et al.  Number-conserving cellular automata I: decidability , 2003, Theor. Comput. Sci..

[5]  J. Myhill.  The converse of Moore’s Garden-of-Eden theorem , 1963 .

[6]  G. A. Hedlund.  Endomorphisms and automorphisms of the shift dynamical system , 1969, Mathematical systems theory.

[7]  M. Mirzakhani,et al.  Introduction to Ergodic theory , 2010 .

[8]  M. Pivato.  Conservation Laws in Cellular Automata , 2001, math/0111014.

[9]  E. F. Moore.  Machine Models of Self-Reproduction , 1962 .

[10]  R. Lathe.  Phd by thesis , 1988, Nature.

[11]  Hendrik B. Geyer,et al.  Journal of Physics A - Mathematical and General, Special Issue. SI Aug 11 2006 ?? Preface , 2006 .

[12]  Jarkko Kari,et al.  On the hierarchy of conservation laws in a cellular automaton , 2011, Natural Computing.

[13]  Vincent Bernardi.  Lois de conservation sur automates cellulaires , 2007 .

[14]  H. Fuks,et al.  Cellular automaton rules conserving the number of active sites , 1997, adap-org/9712003.

[15]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[16]  T. Hattori,et al.  Additive conserved quantities in discrete-time lattice dynamical systems , 1991 .

[17]  J. van Leeuwen,et al.  Theoretical Computer Science , 2003, Lecture Notes in Computer Science.

[18]  C. Preston.  Gibbs States on Countable Sets , 1974 .

[19]  Vicenta Sánchez,et al.  A real-space renormalization approach to the Kubo–Greenwood formula in mirror Fibonacci systems , 2006 .

[20]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[21]  Eric Goles Ch.,et al.  On conservative and monotone one-dimensional cellular automata and their particle representation , 2004, Theor. Comput. Sci..

[22]  Marcus Pivato,et al.  The ergodic theory of cellular automata , 2012, Int. J. Gen. Syst..

[23]  J. Laurie Snell,et al.  Markov Random Fields and Their Applications , 1980 .