The global statistics of return times: return time dimensions versus generalized measure dimensions

We investigate the relations holding among generalized dimensions of invariant measures in dynamical systems and similar quantities defined by the scaling of global averages of powers of return times. Because of a heuristic use of Kac theorem, these latter have been used in place of the former in numerical and experimental investigations; to mark this distinction, we call them return time dimensions. We derive a full set of inequalities linking measure and return time dimensions and we comment on their optimality with the aid of two maps due to von Neumann -- Kakutani and to Gaspard -- Wang. We conjecture the behavior of return time dimensions in a typical system. We only assume ergodicity of the dynamical system under investigation.Comment: Submitted to J. Stat. Phy

Quantum Intermittency in Almost-Periodic Lattice Systems Derived from their Spectral Properties

Hamiltonian tridiagonal matrices characterized by multi-fractal spectral measures in the family of Iterated Function Systems can be constructed by a recursive technique here described. We prove that these Hamiltonians are almost-periodic. They are suited to describe quantum lattice systems with nearest neighbours coupling, as well as chains of linear classical oscillators, and electrical transmission lines. We investigate numerically and theoretically the time dynamics of the systems so constructed. We derive a relation linking the long-time, power-law behaviour of the moments of the position operator, expressed by a scaling function $\beta$ of the moment order $\alpha$, and spectral multi-fractal dimensions, D_q, via $\beta(\alpha) = D_{1-\alpha}$. We show cases in which this relation is exact, and cases where it is only approximate, unveiling the reasons for the discrepancies.Comment: 13 pages, Latex, 6 postscript figures. Accepted for publication in Physica

The Multiparticle Quantum Arnol'd Cat: a test case for the decoherence approach to quantum chaos

A multi-particle extension of the Arnol'd Cat Hamiltonian system is defined and examined. We propose to compute its Alicki-Fannes quantum dynamical entropy, to validate (or disprove) the validity of the decoherence approach to quantum chaos. A first set of numerical experiments is presented and discussed.Comment: To appear in the Journal of the Siberian Federal Universit

Dynamical Systems and Numerical Analysis: the Study of Measures generated by Uncountable I.F.S

Measures generated by Iterated Function Systems composed of uncountably many one--dimensional affine maps are studied. We present numerical techniques as well as rigorous results that establish whether these measures are absolutely or singular continuous.Comment: to appear in Numerical Algorithm

On the Attractor of One-Dimensional Infinite Iterated Function Systems

We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping intervals. Numerical techniques are described to test this conjecture, and a partial rigorous result in this direction is proven.Comment: 10 pages, 4 figure

Quantum Algorithmic Integrability: The Metaphor of Polygonal Billiards

An elementary application of Algorithmic Complexity Theory to the polygonal approximations of curved billiards-integrable and chaotic-unveils the equivalence of this problem to the procedure of quantization of classical systems: the scaling relations for the average complexity of symbolic trajectories are formally the same as those governing the semi-classical limit of quantum systems. Two cases-the circle, and the stadium-are examined in detail, and are presented as paradigms.Comment: 11 pages, 5 figure