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space entropy of elt. cell. automata
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Posted by: Steve Seif
Are there estimates of the ``topological entropy'' of the elementary cellular automata?
Here, for an elementary cellular automaton CA, h(CA) (the topological entropy of CA) is given by
lim_{n to infty} log( B_n(CA))/n,
where B_n(CA) is the number of tuples in
{0,1}^n that are image under CA of some tuple in {0,1}^n+2.
For example, if CA is surjective then
B_n(CA) = {0,1}^n, for all positive integers n; if CA is not surjective, then there is some n such that B_n(CA) is properly contained in {0,1}^n. And, if CA is surjective, then h(CA) = log 2.
It is known that if CA is not surjective, then h(CA) < log 2.
Topological entropy does not take into account evolution over time. For this reason perhaps, it is of less interest to
many who work in cellular automata. But, it is of great interest of me. Any direction on this matter would be greatly appreciated.
Sincerely, Steve Seif, Math Dep't, University of Louisville, swseif01@louisville.edu
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