468 research outputs found
The topological strong spatial mixing property and new conditions for pressure approximation
In the context of stationary nearest-neighbour Gibbs measures
satisfying strong spatial mixing, we present a new combinatorial
condition (the topological strong spatial mixing property (TSSM)) on the
support of sufficient for having an efficient approximation algorithm for
topological pressure. We establish many useful properties of TSSM for studying
strong spatial mixing on systems with hard constraints. We also show that TSSM
is, in fact, necessary for strong spatial mixing to hold at high rate. Part of
this work is an extension of results obtained by D. Gamarnik and D. Katz
(2009), and B. Marcus and R. Pavlov (2013), who gave a special representation
of topological pressure in terms of conditional probabilities.Comment: 40 pages, 8 figures. arXiv admin note: text overlap with
arXiv:1309.1873 by other author
Computability and dynamical systems
In this paper we explore results that establish a link between dynamical
systems and computability theory (not numerical analysis). In the last few decades,
computers have increasingly been used as simulation tools for gaining insight into
dynamical behavior. However, due to the presence of errors inherent in such numerical
simulations, with few exceptions, computers have not been used for the
nobler task of proving mathematical results. Nevertheless, there have been some recent
developments in the latter direction. Here we introduce some of the ideas and
techniques used so far, and suggest some lines of research for further work on this
fascinating topic
Information complexity is computable
The information complexity of a function is the minimum amount of
information Alice and Bob need to exchange to compute the function . In this
paper we provide an algorithm for approximating the information complexity of
an arbitrary function to within any additive error , thus
resolving an open question as to whether information complexity is computable.
In the process, we give the first explicit upper bound on the rate of
convergence of the information complexity of when restricted to -bit
protocols to the (unrestricted) information complexity of .Comment: 30 page
The categorical basis of dynamical entropy
Many different branches of theoretical and applied mathematics require a
quantifiable notion of complexity. One such circumstance is a topological
dynamical system - which involves a continuous self-map on a metric space.
There are many notions of complexity one can assign to the repeated iterations
of the map. One of the foundational discoveries of dynamical systems theory is
that these have a common limit, known as the topological entropy of the system.
We present a category-theoretic view of topological dynamical entropy, which
reveals that the common limit is a consequence of the structural assumptions on
these notions. One of the key tools developed is that of a qualifying pair of
functors, which ensure a limit preserving property in a manner similar to the
sandwiching theorem from Real Analysis. It is shown that the diameter and
Lebesgue number of open covers of a compact space, form a qualifying pair of
functors. The various notions of complexity are expressed as functors, and
natural transformations between these functors lead to their joint convergence
to the common limit
Ergodic theory on coded shift spaces
We study ergodic-theoretic properties of coded shift spaces. A coded shift
space is defined as a closure of all bi-infinite concatenations of words from a
fixed countable generating set. We derive sufficient conditions for the
uniqueness of measures of maximal entropy and equilibrium states of H\"{o}lder
continuous potentials based on the partition of the coded shift into its
sequential set (sequences that are concatenations of generating words) and its
residual set (sequences added under the closure). In this case we provide a
simple explicit description of the measure of maximal entropy. We also obtain
flexibility results for the entropy on the sequential and residual set.
Finally, we prove a local structure theorem for intrinsically ergodic coded
shift spaces which shows that our results apply to a larger class of coded
shift spaces compared to previous works by Climenhaga, Climenhaga and Thompson,
and Pavlov.Comment: 43 page
Computability Theory (hybrid meeting)
Over the last decade computability theory has seen many new and
fascinating developments that have linked the subject much closer
to other mathematical disciplines inside and outside of logic.
This includes, for instance, work on enumeration degrees that
has revealed deep and surprising relations to general topology,
the work on algorithmic randomness that is closely tied to
symbolic dynamics and geometric measure theory.
Inside logic there are connections to model theory, set theory, effective descriptive
set theory, computable analysis and reverse mathematics.
In some of these cases the bridges to seemingly distant mathematical fields
have yielded completely new proofs or even solutions of open problems
in the respective fields. Thus, over the last decade, computability theory
has formed vibrant and beneficial interactions with other mathematical
fields.
The goal of this workshop was to bring together researchers representing
different aspects of computability theory to discuss recent advances, and to
stimulate future work
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