118 research outputs found
Conjugacy of one-dimensional one-sided cellular automata is undecidable
Two cellular automata are strongly conjugate if there exists a
shift-commuting conjugacy between them. We prove that the following two sets of
pairs of one-dimensional one-sided cellular automata over a full shift
are recursively inseparable: (i) pairs where has strictly larger
topological entropy than , and (ii) pairs that are strongly conjugate and
have zero topological entropy.
Because there is no factor map from a lower entropy system to a higher
entropy one, and there is no embedding of a higher entropy system into a lower
entropy system, we also get as corollaries that the following decision problems
are undecidable: Given two one-dimensional one-sided cellular automata and
over a full shift: Are and conjugate? Is a factor of ? Is
a subsystem of ? All of these are undecidable in both strong and weak
variants (whether the homomorphism is required to commute with the shift or
not, respectively). It also immediately follows that these results hold for
one-dimensional two-sided cellular automata.Comment: 12 pages, 2 figures, accepted for SOFSEM 201
Shifts of finite type with nearly full entropy
For any fixed alphabet A, the maximum topological entropy of a Z^d subshift
with alphabet A is obviously log |A|. We study the class of nearest neighbor
Z^d shifts of finite type which have topological entropy very close to this
maximum, and show that they have many useful properties. Specifically, we prove
that for any d, there exists beta_d such that for any nearest neighbor Z^d
shift of finite type X with alphabet A for which log |A| - h(X) < beta_d, X has
a unique measure of maximal entropy. Our values of beta_d decay polynomially
(like O(d^(-17))), and we prove that the sequence must decay at least
polynomially (like d^(-0.25+o(1))). We also show some other desirable
properties for such X, for instance that the topological entropy of X is
computable and that the unique m.m.e. is isomorphic to a Bernoulli measure.
Though there are other sufficient conditions in the literature which guarantee
a unique measure of maximal entropy for Z^d shifts of finite type, this is (to
our knowledge) the first such condition which makes no reference to the
specific adjacency rules of individual letters of the alphabet.Comment: 33 pages, accepted by Proceedings of the London Mathematical Societ
On the zero-temperature limit of Gibbs states
We exhibit Lipschitz (and hence H\"older) potentials on the full shift
such that the associated Gibbs measures fail to converge
as the temperature goes to zero. Thus there are "exponentially decaying"
interactions on the configuration space for which the
zero-temperature limit of the associated Gibbs measures does not exist. In
higher dimension, namely on the configuration space ,
, we show that this non-convergence behavior can occur for finite-range
interactions, that is, for locally constant potentials.Comment: The statement of Theorem 1.2 is more accurate and some new comment
follow i
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