Resolvent estimates for operators belonging to exponential classes

For $a,\alpha>0$ let $E(a,\alpha)$ be the set of all compact operators $A$ on a separable Hilbert space such that $s_n(A)=O(\exp(-an^\alpha))$, where $s_n(A)$ denotes the $n$-th singular number of $A$. We provide upper bounds for the norm of the resolvent $(zI-A)^{-1}$ of $A$ in terms of a quantity describing the departure from normality of $A$ and the distance of $z$ to the spectrum of $A$. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in $E(a,\alpha)$.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 $h$. Using ideas from dynamical systems and functional analysis, we show that the free energy $f$ in the vicinity of the second-order phase transition is given, exactly, by $$f \sim \frac t{\log t}-\frac1{2} \frac{h^2}t \quad \text{for} \quad h^2\ll t \ll 1 .$$ Here $t=\lambda_{G}\log(2)(1-\frac{\beta}{\beta_c})$ is a reduced temperature, so that the deviation from the critical point is scaled by the Lyapunov exponent of the Gauss map, $\lambda_G$. It follows that $\lambda_G$ 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