811 research outputs found
On Sushchansky p-groups
We study Sushchansky p-groups. We recall the original definition and
translate it into the language of automata groups. The original actions of
Sushchansky groups on p-ary tree are not level-transitive and we describe their
orbit trees. This allows us to simplify the definition and prove that these
groups admit faithful level-transitive actions on the same tree. Certain branch
structures in their self-similar closures are established. We provide the
connection with, so-called, G groups that shows that all Sushchansky groups
have intermediate growth and allows to obtain an upper bound on their period
growth functions.Comment: 14 pages, 3 figure
No Tits alternative for cellular automata
We show that the automorphism group of a one-dimensional full shift (the
group of reversible cellular automata) does not satisfy the Tits alternative.
That is, we construct a finitely-generated subgroup which is not virtually
solvable yet does not contain a free group on two generators. We give
constructions both in the two-sided case (spatially acting group Z) and the
one-sided case (spatially acting monoid N, alphabet size at least eight).
Lack of Tits alternative follows for several groups of symbolic (dynamical)
origin: automorphism groups of two-sided one-dimensional uncountable sofic
shifts, automorphism groups of multidimensional subshifts of finite type with
positive entropy and dense minimal points, automorphism groups of full shifts
over non-periodic groups, and the mapping class groups of two-sided
one-dimensional transitive SFTs. We also show that the classical Tits
alternative applies to one-dimensional (multi-track) reversible linear cellular
automata over a finite field.
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A Garden of Eden theorem for Anosov diffeomorphisms on tori
Let be an Anosov diffeomorphism of the -dimensional torus
and a continuous self-mapping of
commuting with . We prove that is surjective if and only if the
restriction of to each homoclinicity class of is injective.Comment: 9 page
Iterated Monodromy Groups of Quadratic Polynomials, I
We describe the iterated monodromy groups associated with post-critically
finite quadratic polynomials, and explicit their connection to the `kneading
sequence' of the polynomial.
We then give recursive presentations by generators and relations for these
groups, and study some of their properties, like torsion and `branchness'.Comment: 18 pages, 3 EPS figure
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