36 research outputs found
Random subshifts of finite type
Let be an irreducible shift of finite type (SFT) of positive entropy, and
let be its set of words of length . Define a random subset
of by independently choosing each word from with some
probability . Let be the (random) SFT built from the set
. For each and tending to infinity, we compute
the limit of the likelihood that is empty, as well as the limiting
distribution of entropy for . For near 1 and tending
to infinity, we show that the likelihood that contains a unique
irreducible component of positive entropy converges exponentially to 1. These
results are obtained by studying certain sequences of random directed graphs.
This version of "random SFT" differs significantly from a previous notion by
the same name, which has appeared in the context of random dynamical systems
and bundled dynamical systems.Comment: Published in at http://dx.doi.org/10.1214/10-AOP636 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Orders of accumulation of entropy
For a continuous map of a compact metrizable space with finite
topological entropy, the order of accumulation of entropy of is a countable
ordinal that arises in the context of entropy structure and symbolic
extensions. We show that every countable ordinal is realized as the order of
accumulation of some dynamical system. Our proof relies on functional analysis
of metrizable Choquet simplices and a realization theorem of Downarowicz and
Serafin. Further, if is a metrizable Choquet simplex, we bound the ordinals
that appear as the order of accumulation of entropy of a dynamical system whose
simplex of invariant measures is affinely homeomorphic to . These bounds are
given in terms of the Cantor-Bendixson rank of \overline{\ex(M)}, the closure
of the extreme points of , and the relative Cantor-Bendixson rank of
\overline{\ex(M)} with respect to \ex(M). We also address the optimality of
these bounds.Comment: 48 page
Orders of accumulation of entropy and random subshifts of finite type
For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on the functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if M is a metrizable Choquet simplex, we bound the ordinals that appear as the order of accumulation of entropy of a dynamical system whose simplex of invariant measures is affinely homeomorphic to M. These bounds are given in terms of the Cantor-Bendixson rank of F, the closure of the extreme points of M, and the relative Cantor-Bendixson rank of F with respect to the extreme points of M. We address the optimality of these bounds.
Given any compact manifold M and any countable ordinal alpha, we construct a continuous, surjective self-map of M having order of accumulation of entropy alpha. If the dimension of M is at least 2, then the map can be chosen to be a homeomorphism. The realization theorem of Downarowicz and Serafin produces dynamical systems on the Cantor set; by contrast, our constructions work on any manifold and provide a more direct dynamical method of obtaining systems with prescribed entropy properties.
Next we consider random subshifts of finite type. Let X be an irreducible shift of finite type (SFT) of positive entropy with its set of words of length n denoted B_n(X). Define a random subset E of B_n(X) by independently choosing each word from B_n(X) with some probability alpha. Let X_E be the (random) SFT built from the set E. For each alpha in [0,1] and n tending to infinity, we compute the limit of the likelihood that X_E; is empty, as well as the limiting distribution of entropy for X_E. For alpha near 1 and n tending to infinity, we show that the likelihood that X_E contains a unique irreducible component of positive entropy converges exponentially to 1
Consistency of maximum likelihood estimation for some dynamical systems
We consider the asymptotic consistency of maximum likelihood parameter
estimation for dynamical systems observed with noise. Under suitable conditions
on the dynamical systems and the observations, we show that maximum likelihood
parameter estimation is consistent. Our proof involves ideas from both
information theory and dynamical systems. Furthermore, we show how some
well-studied properties of dynamical systems imply the general statistical
properties related to maximum likelihood estimation. Finally, we exhibit
classical families of dynamical systems for which maximum likelihood estimation
is consistent. Examples include shifts of finite type with Gibbs measures and
Axiom A attractors with SRB measures.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1259 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org