Geometric phases in quantum control disturbed by classical stochastic processes

We describe the geometric (Berry) phases arising when some quantum systems are driven by control classical parameters but also by outer classical stochastic processes (as for example classical noises). The total geometric phase is then divided into an usual geometric phase associated with the control parameters and a second geometric phase associated with the stochastic processes. The geometric structure in which these geometric phases take place is a composite bundle (and not an usual principal bundle), which is explicitely built in this paper. We explain why the composite bundle structure is the more natural framework to study this problem. Finally we treat a very simple example of a two level atom driven by a phase modulated laser field with a phase instability described by a gaussian white noise. In particular we compute the average geometric phase issued from the noise

Natural extensions and entropy of α\alpha-continued fractions

We construct a natural extension for each of Nakada's $\alpha$-continued fractions and show the continuity as a function of $\alpha$ of both the entropy and the measure of the natural extension domain with respect to the density function $(1+xy)^{-2}$. In particular, we show that, for all $0 < \alpha \le 1$, the product of the entropy with the measure of the domain equals $\pi^2/6$. As a key step, we give the explicit relationship between the $\alpha$-expansion of $\alpha-1$ and of $\alpha$

The mathematical research of William Parry FRS

In this article we survey the mathematical research of the late William (Bill) Parry, FRS

Separation of trajectories and its Relation to Entropy for Intermittent Systems with a Zero Lyapunov exponent

One dimensional intermittent maps with stretched exponential separation of nearby trajectories are considered. When time goes infinity the standard Lyapunov exponent is zero. We investigate the distribution of $\lambda_{\alpha}= \sum_{i=0}^{t-1} \ln \left| M'(x_i) \right|/t^{\alpha}$, where $\alpha$ is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points. The mean of $\lambda_{\alpha}$ is determined by the infinite invariant density. Using semi analytical arguments we calculate the infinite invariant density for the Pomeau-Manneville map, and with it obtain excellent agreement between numerical simulation and theory. We show that \alpha \left is equal to Krengel's entropy and to the complexity calculated by the Lempel-Ziv compression algorithm. This generalized Pesin's identity shows that \left and Krengel's entropy are the natural generalizations of usual Lyapunov exponent and entropy for these systems.Comment: 12 pages, 10 figure

Pesin-type relation for subexponential instability

We address here the problem of extending the Pesin relation among positive Lyapunov exponents and the Kolmogorov-Sinai entropy to the case of dynamical systems exhibiting subexponential instabilities. By using a recent rigorous result due to Zweim\"uller, we show that the usual Pesin relation can be extended straightforwardly for weakly chaotic one-dimensional systems of the Pomeau-Manneville type, provided one introduces a convenient subexponential generalization of the Kolmogorov-Sinai entropy. We show, furthermore, that Zweim\"uller's result provides an efficient prescription for the evaluation of the algorithm complexity for such systems. Our results are confirmed by exhaustive numerical simulations. We also point out and correct a misleading extension of the Pesin relation based on the Krengel entropy that has appeared recently in the literature.Comment: 10 pages, 4 figures. Final version to appear in Journal of Statistical Mechanics (JSTAT

The entropy of alpha-continued fractions: numerical results

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.Comment: 33 pages, 14 figure

Exotic Differentiable Structures and General Relativity

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (``fake'' or ``exotic'') differentiable structures on topologically simple manifolds such as $S^7$, \R and $S^3\times {\bf R^1}.$ Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.Comment: 11 pages, LaTe

On conformal measures and harmonic functions for group extensions

We prove a Perron-Frobenius-Ruelle theorem for group extensions of topological Markov chains based on a construction of $\sigma$-finite conformal measures and give applications to the construction of harmonic functions.Comment: To appear in Proceedings of "New Trends in Onedimensional Dynamics, celebrating the 70th birthday of Welington de Melo