53 research outputs found
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
of the moment order , and spectral multi-fractal dimensions,
D_q, via . 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
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