Entropy of semiclassical measures for nonpositively curved surfaces

We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. We show that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound. We follow the same main strategy as in the Anosov case (arXiv:0809.0230). We focus on the main differences and refer the reader to (arXiv:0809.0230) for the details of analogous lemmas.Comment: 20 pages. This note provides a detailed proof of a result announced in appendix A of a previous work (arXiv:0809.0230, version 2

Delocalization of slowly damped eigenmodes on Anosov manifolds

We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with spectral parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a "strip" of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.Comment: 28 pages; compared with version 1, minor modifications, add two reference

Radon--Nikodym representations of Cuntz--Krieger algebras and Lyapunov spectra for KMS states

We study relations between $(H,\beta)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-\beta H}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,\beta)$--KMS states and eigenmeasures of $\mathcal{L}_{-\beta H}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--isomorphic to the given Cuntz--Krieger algebra. Finally, we apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups $G$ which may have parabolic elements. We show that for the Cuntz--Krieger algebra arising from $G$ there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen--Series map associated with $G$. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of $G$. If $G$ has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with $G$.Comment: 30 pages, minor changes in the proofs of Theorem 3.9 and Fact

Experiences with Mycobacterium leprae soluble antigens in a leprosy endemic population

Rees and Convit antigens prepared from armadillo-derived Mycobacterium leprae were used for skin testing in two leprosy endemic villages to understand their use in the epidemiology of leprosy. In all, 2602 individuals comprising 202 patients with leprosy detected in a prevalence survey, 476 household contacts and 1924 persons residing in non-case households were tested with two antigens. There was a strong and positive correlation ( r = 0.85) between reactions to the Rees and Convit antigens. The distribution of reactions was bimodal and considering reactions of 12 mm or more as ‘positive’, the positivity rate steeply increased with the increase in age. However. the distributions of reactions to these antigens in patients with leprosy. their household contacts and persons living in non-case households were very similar. These results indicate that Rees and Convit antigens are not useful in the identification of M. leprae infection or in the confirmation of leprosy diagnosis in a leprosy endemic population with a high prevalence of nonspecific sensitivity

Semiclassical measures and the Schroedinger flow on Riemannian manifolds

In this article we study limits of Wigner distributions (the so-called semiclassical measures) corresponding to sequences of solutions to the semiclassical Schroedinger equation at times scales $\alpha_{h}$ tending to infinity as the semiclassical parameter $h$ tends to zero (when $\alpha _{h}=1/h$ this is equivalent to consider solutions to the non-semiclassical Schreodinger equation). Some general results are presented, among which a weak version of Egorov's theorem that holds in this setting. A complete characterization is given for the Euclidean space and Zoll manifolds (that is, manifolds with periodic geodesic flow) via averaging formulae relating the semiclassical measures corresponding to the evolution to those of the initial states. The case of the flat torus is also addressed; it is shown that non-classical behavior may occur when energy concentrates on resonant frequencies. Moreover, we present an example showing that the semiclassical measures associated to a sequence of states no longer determines those of their evolutions. Finally, some results concerning the equation with a potential are presented.Comment: 18 pages; Theorems 1,2 extendend to deal with arbitrary time-scales; references adde

Quantisations of piecewise affine maps on the torus and their quantum limits

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits. The maps we quantise are obtained by cutting and stacking

Tuberculosis prevalence survey in Kashmir valley

A tuberculosis prevalence survey was conducted in about 18,000 persons in Kashmir valley situated about 1650 m above the mean sea level. All persons were tested with 3 IU of PPD-S and 10 units of PPD-B. Persons aged 5 yr and above were X-rayed (70 mm X-ray), and from such persons whose photofluorograms were interpreted as abnormal two specimens of sputum were collected and bacteriologically examined. In addition, a large X-ray of the chest was taken for children aged 0-4 yr who had reactions of 10 mm or more to PPD-S. They were then clinically examined by a paediatrician, taking into account all available data, for evaluation for any evidence of tuberculosis. The results of the survey showed that the prevalence of non-specific sensitivity (59%) in the Kashmir valley is significant. The prevalence of tuberculous infection was 38 per cent. The prevalence of culture positive tuberculous patients (3 per 1000) and that of abacillary X-ray positive patients (14 per 1000) were found to be similar in the two sexes contrary to the usual experience of a higher prevalence among males. Results from studies of phage typing, susceptibility to thiophen-2-carbonic acid hydrazide (TCH) and virulence in the guineapig of strains obtained from patients diagnosed in the survey showed that most of these strains belonged to phage type A, were resistant to TCH and were not of low virulence. Regional variations in the prevalence rates were seen, the- problem of tuberculosis appeared to he more in the Baramulla district as compared to Srinagar and Anantnag districts. A comparison of results obtained from the present survey with those obtained from the BCG trial in Chingleput (Tamil Nadu) revealed that the tuberculosis situation in the two areas was quite different

Anatomy of quantum chaotic eigenstates

The eigenfunctions of quantized chaotic systems cannot be described by explicit formulas, even approximate ones. This survey summarizes (selected) analytical approaches used to describe these eigenstates, in the semiclassical limit. The levels of description are macroscopic (one wants to understand the quantum averages of smooth observables), and microscopic (one wants informations on maxima of eigenfunctions, "scars" of periodic orbits, structure of the nodal sets and domains, local correlations), and often focusses on statistical results. Various models of "random wavefunctions" have been introduced to understand these statistical properties, with usually good agreement with the numerical data. We also discuss some specific systems (like arithmetic ones) which depart from these random models.Comment: Corrected typos, added a few references and updated some result