24 research outputs found
Resolvent estimates for operators belonging to exponential classes
For let be the set of all compact operators on
a separable Hilbert space such that , where
denotes the -th singular number of . We provide upper bounds for
the norm of the resolvent of in terms of a quantity
describing the departure from normality of and the distance of to the
spectrum of . As a consequence we obtain upper bounds for the Hausdorff
distance of the spectra of two operators in .Comment: AMS-LaTeX, 20 page
Estimates for norms of resolvents and an application to the per- turbation of spectra
Abstract. Let A belong to the Schatten-von Neumann ideal Sp for 0 < p < ∞. We give an upper bound for the operator norm of the resolvent (zI − A) −1 of A in terms of the departure from normality of A and the distance of z to the spectrum of A. As an application we provide an upper bound for the Hausdorff distance of the spectra of two operators belonging to Sp
Lower bounds for the Ruelle spectrum of analytic expanding circle maps
We prove that there exists a dense set of analytic expanding maps of the
circle for which the Ruelle eigenvalues enjoy exponential lower bounds. The
proof combines potential theoretic techniques and explicit calculations for the
spectrum of expanding Blaschke products.Comment: 21 pages, 1 figur
Entropy continuity for interval maps with holes
We study the dependence of the topological entropy of piecewise monotonic
maps with holes under perturbations, for example sliding a hole of fixed size
at uniform speed or expanding a hole with uniform expansion. We show that under
suitable conditions the topological entropy varies locally Hoelder continuously
with the local Hoelder exponent depending itself on the value of the
topological entropy.Comment: 23 pages; section 6 has been considerably simplified following
suggestions of a referee; to appear in Ergodic Theory and Dynamical System
Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point
We consider the Farey fraction spin chain in an external field . Using
ideas from dynamical systems and functional analysis, we show that the free
energy in the vicinity of the second-order phase transition is given,
exactly, by
Here is a reduced
temperature, so that the deviation from the critical point is scaled by the
Lyapunov exponent of the Gauss map, . It follows that
determines the amplitude of both the specific heat and susceptibility
singularities. To our knowledge, there is only one other microscopically
defined interacting model for which the free energy near a phase transition is
known as a function of two variables.
Our results confirm what was found previously with a cluster approximation,
and show that a clustering mechanism is in fact responsible for the transition.
However, the results disagree in part with a renormalisation group treatment