806 research outputs found
On algebraic cellular automata
We investigate some general properties of algebraic cellular automata, i.e.,
cellular automata over groups whose alphabets are affine algebraic sets and
which are locally defined by regular maps. When the ground field is assumed to
be uncountable and algebraically closed, we prove that such cellular automata
always have a closed image with respect to the prodiscrete topology on the
space of configurations and that they are reversible as soon as they are
bijective
On surjunctive monoids
A monoid is called surjunctive if every injective cellular automata with
finite alphabet over is surjective. We show that all finite monoids, all
finitely generated commutative monoids, all cancellative commutative monoids,
all residually finite monoids, all finitely generated linear monoids, and all
cancellative one-sided amenable monoids are surjunctive. We also prove that
every limit of marked surjunctive monoids is itself surjunctive. On the other
hand, we show that the bicyclic monoid and, more generally, all monoids
containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive
Expansive actions of countable amenable groups, homoclinic pairs, and the Myhill property
Let be a compact metrizable space equipped with a continuous action of a
countable amenable group . Suppose that the dynamical system is
expansive and is the quotient by a uniformly bounded-to-one factor map of a
strongly irreducible subshift. Let be a continuous map
commuting with the action of . We prove that if there is no pair of distinct
-homoclinic points in having the same image under , then is
surjective.Comment: arXiv admin note: text overlap with arXiv:1506.0694
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