129 research outputs found
The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
Equivalence of relative Gibbs and relative equilibrium measures for actions of countable amenable groups
We formulate and prove a very general relative version of the
Dobrushin-Lanford-Ruelle theorem which gives conditions on constraints of
configuration spaces over a finite alphabet such that for every absolutely
summable relative interaction, every translation-invariant relative Gibbs
measure is a relative equilibrium measure and vice versa. Neither implication
is true without some assumption on the space of configurations. We note that
the usual finite type condition can be relaxed to a much more general class of
constraints. By "relative" we mean that both the interaction and the set of
allowed configurations are determined by a random environment. The result
includes many special cases that are well known. We give several applications
including (1) Gibbsian properties of measures that maximize pressure among all
those that project to a given measure via a topological factor map from one
symbolic system to another; (2) Gibbsian properties of equilibrium measures for
group shifts defined on arbitrary countable amenable groups; (3) A Gibbsian
characterization of equilibrium measures in terms of equilibrium condition on
lattice slices rather than on finite sets; (4) A relative extension of a
theorem of Meyerovitch, who proved a version of the Lanford--Ruelle theorem
which shows that every equilibrium measure on an arbitrary subshift satisfies a
Gibbsian property on interchangeable patterns.Comment: 37 pages and 3 beautiful figure
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
Defect Particle Kinematics in One-Dimensional Cellular Automata
Let A^Z be the Cantor space of bi-infinite sequences in a finite alphabet A,
and let sigma be the shift map on A^Z. A `cellular automaton' is a continuous,
sigma-commuting self-map Phi of A^Z, and a `Phi-invariant subshift' is a
closed, (Phi,sigma)-invariant subset X of A^Z. Suppose x is a sequence in A^Z
which is X-admissible everywhere except for some small region we call a
`defect'. It has been empirically observed that such defects persist under
iteration of Phi, and often propagate like `particles'. We characterize the
motion of these particles, and show that it falls into several regimes, ranging
from simple deterministic motion, to generalized random walks, to complex
motion emulating Turing machines or pushdown automata. One consequence is that
some questions about defect behaviour are formally undecidable.Comment: 37 pages, 9 figures, 3 table
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