Invariance entropy, quasi-stationary measures and control sets

For control systems in discrete time, this paper discusses measure-theoretic invariance entropy for a subset Q of the state space with respect to a quasi-stationary measure obtained by endowing the control range with a probability measure. The main results show that this entropy is invariant under measurable transformations and that it is already determined by certain subsets of Q which are characterized by controllability properties.Comment: 30 page

Entropy of nonautonomous dynamical systems

Different notions of entropy play a fundamental role in the classical theory of dynamical systems. Unlike many other concepts used to analyze autonomous dynamics, both measure-theoretic and topological entropy can be extended quite naturally to discrete-time nonautonomous dynamical systems given in the process formulation. This paper provides an overview of the author's work on this subject. Also an example is presented that has not appeared before in the literature

Spreading lengths of Hermite polynomials

The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted L^q-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, it is computationally proved the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments mentioned previously are posed.Comment: 16 pages, 4 figures. Journal of Computational and Applied Mathematics (2009), doi:10.1016/j.cam.2009.09.04