9,970 research outputs found
Entropy and drift for word metric on relatively hyperbolic groups
We are interested in the Guivarc'h inequality for admissible random walks on
finitely generated relatively hyperbolic groups, endowed with a word metric. We
show that for random walks with finite super-exponential moment, if this
inequality is an equality, then the Green distance is roughly similar to the
word distance, generalizing results of Blach{\`e}re, Ha{\"i}ssinsky and Mathieu
for hyperbolic groups [4]. Our main application is for relatively hyperbolic
groups with respect to virtually abelian subgroups of rank at least 2. We show
that for such groups, the Guivarc'h inequality with respect to a word distance
and a finitely supported random walk is always strict
An entropy for groups of intermediate growth
One of the few accepted dynamical foundations of non-additive
"non-extensive") statistical mechanics is that the choice of the appropriate
entropy functional describing a system with many degrees of freedom should
reflect the rate of growth of its configuration or phase space volume. We
present an example of a group, as a metric space, that may be used as the phase
space of a system whose ergodic behavior is statistically described by the
recently proposed -entropy. This entropy is a one-parameter variation
of the Boltzmann/Gibbs/Shannon functional and is quite different, in form, from
the power-law entropies that have been recently studied. We use the first
Grigorchuk group for our purposes. We comment on the connections of the above
construction with the conjectured evolution of the underlying system in phase
space.Comment: 19 pages, No figures, LaTeX2e. Version 2: change of affiliation,
addition of acknowledgement. Accepted for publication to "Advances in
Mathematical Physics
- …