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Some properties of cellular automata with equicontinuity points
We investigate topological and ergodic properties of cellular automata having
equicontinuity points. In this class surjectivity on a transitive SFT implies
existence of a dense set of periodic points. Our main result is that under the
action of such an automaton any shift ergodic measure converges in Cesaro Mean
Formal Languages in Dynamical Systems
We treat here the interrelation between formal languages and those dynamical
systems that can be described by cellular automata (CA). There is a well-known
injective map which identifies any CA-invariant subshift with a central formal
language. However, in the special case of a symbolic dynamics, i.e. where the
CA is just the shift map, one gets a stronger result: the identification map
can be extended to a functor between the categories of symbolic dynamics and
formal languages. This functor additionally maps topological conjugacies
between subshifts to empty-string-limited generalized sequential machines
between languages. If the periodic points form a dense set, a case which arises
in a commonly used notion of chaotic dynamics, then an even more natural map to
assign a formal language to a subshift is offered. This map extends to a
functor, too. The Chomsky hierarchy measuring the complexity of formal
languages can be transferred via either of these functors from formal languages
to symbolic dynamics and proves to be a conjugacy invariant there. In this way
it acquires a dynamical meaning. After reviewing some results of the complexity
of CA-invariant subshifts, special attention is given to a new kind of
invariant subshift: the trapped set, which originates from the theory of
chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe
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