1,605 research outputs found
Equilibrium states and invariant measures for random dynamical systems
Random dynamical systems with countably many maps which admit countable
Markov partitions on complete metric spaces such that the resulting Markov
systems are uniformly continuous and contractive are considered. A
non-degeneracy and a consistency conditions for such systems, which admit some
proper Markov partitions of connected spaces, are introduced, and further
sufficient conditions for them are provided. It is shown that every uniformly
continuous Markov system associated with a continuous random dynamical system
is consistent if it has a dominating Markov chain. A necessary and sufficient
condition for the existence of an invariant Borel probability measure for such
a non-degenerate system with a dominating Markov chain and a finite (16) is
given. The condition is also sufficient if the non-degeneracy is weakened with
the consistency condition. A further sufficient condition for the existence of
an invariant measure for such a consistent system which involves only the
properties of the dominating Markov chain is provided. In particular, it
implies that every such a consistent system with a finite Markov partition and
a finite (16) has an invariant Borel probability measure. A bijective map
between these measures and equilibrium states associated with such a system is
established in the non-degenerate case. Some properties of the map and the
measures are given.Comment: The article is published in DCDS-A, but without the 3rd paragraph on
page 4 (the complete removal of the paragraph became the condition for the
publication in the DCDS-A after the reviewer ran out of the citation
suggestions collected in the paragraph
Threshold graph limits and random threshold graphs
We study the limit theory of large threshold graphs and apply this to a
variety of models for random threshold graphs. The results give a nice set of
examples for the emerging theory of graph limits.Comment: 47 pages, 8 figure
The Solecki submeasures and densities on groups
We introduce the Solecki submeasure and its left and right modifications on a group , and study the
interplay between the Solecki submeasure and the Haar measure on compact
topological groups. Also we show that the right Solecki density on a countable
amenable group coincides with the upper Banach density which allows us to
generalize some fundamental results of Bogoliuboff, Folner, Cotlar and
Ricabarra, Ellis and Keynes about difference sets and Jin, Beiglbock, Bergelson
and Fish about the sumsets to the class of all amenable groups.Comment: 34 page
The structure of unicellular maps, and a connection between maps of positive genus and planar labelled trees
A unicellular map is a map which has only one face. We give a bijection
between a dominant subset of rooted unicellular maps of fixed genus and a set
of rooted plane trees with distinguished vertices. The bijection applies as
well to the case of labelled unicellular maps, which are related to all rooted
maps by Marcus and Schaeffer's bijection.
This gives an immediate derivation of the asymptotic number of unicellular
maps of given genus, and a simple bijective proof of a formula of Lehman and
Walsh on the number of triangulations with one vertex. From the labelled case,
we deduce an expression of the asymptotic number of maps of genus g with n
edges involving the ISE random measure, and an explicit characterization of the
limiting profile and radius of random bipartite quadrangulations of genus g in
terms of the ISE.Comment: 27pages, 6 figures, to appear in PTRF. Version 2 includes corrections
from referee report in sections 6-
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