On Quantum Iterated Function Systems

Quantum Iterated Function System on a complex projective space is defined by a family of linear operators on a complex Hilbert space. The operators define both the maps and their probabilities by one algebraic formula. Examples with conformal maps (relativistic boosts) on the Bloch sphere are discussed.Comment: Latex, 12 pages, 3 figures. Added plot of numerical estimate of the averaged contraction parameter fro quantum octahedron over the whole range of the fuzziness parameter. Added a theorem and proof of the uniqueness of the invariant measure. At the very end added subsection on "open problems

Poisson-to-Wigner crossover transition in the nearest-neighbor spacing statistics of random points on fractals

We show that the nearest-neighbor spacing distribution for a model that consists of random points uniformly distributed on a self-similar fractal is the Brody distribution of random matrix theory. In the usual context of Hamiltonian systems, the Brody parameter does not have a definite physical meaning, but in the model considered here, the Brody parameter is actually the fractal dimension. Exploiting this result, we introduce a new model for a crossover transition between Poisson and Wigner statistics: random points on a continuous family of self-similar curves with fractal dimension between 1 and 2. The implications to quantum chaos are discussed, and a connection to conservative classical chaos is introduced.Comment: Low-resolution figure is included here. Full resolution image available (upon request) from the author

Inter-Intra Molecular Dynamics as an Iterated Function System

The dynamics of units (molecules) with slowly relaxing internal states is studied as an iterated function system (IFS) for the situation common in e.g. biological systems where these units are subjected to frequent collisional interactions. It is found that an increase in the collision frequency leads to successive discrete states that can be analyzed as partial steps to form a Cantor set. By considering the interactions among the units, a self-consistent IFS is derived, which leads to the formation and stabilization of multiple such discrete states. The relevance of the results to dynamical multiple states in biomolecules in crowded conditions is discussed.Comment: 7 pages, 7 figures. submitted to Europhysics Letter

Julia Sets of Orthogonal Polynomials

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {P n } to properties of the support. More precisely we relate the Julia set of P n to the outer boundary of the support, the filled Julia set to the polynomial convex hull K of the support, and the Green’s function associated with P n to the Green’s function for the complement of K

On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions

We consider the "Mandelbrot set" $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ is connected. We show that a non-trivial portion of $M$ near the imaginary axis is contained in the closure of its interior (it is conjectured that all non-real points of $M$ are in the closure of the set of interior points of $M$). Next we turn to the attractors $A_\lambda$ themselves and to natural measures $\nu_\lambda$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures $\nu_\lambda$. Next we investigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$ in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on the absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greater than $\sqrt{1/2}$.Comment: 22 pages, 5 figure

Relative age effect in European professional football. Analysis by position

The relative age effect (RAE) in sport consists in the lower presence of athletes born in the months furthest from the cut off date established by the competitive system, which normally coincides with the last months of the year. The purpose of this study is to assess if there is an RAE in European professional football, where the current cut off age for training in the different categories is based on the calendar year; and its analysis by position on the playing field. The analysis has included all the footballers playing in the league championships in the United Kingdom, Italy, Germany, France and Spain (N= 2763). The results of our study confirm an RAE in professional football in Italy, France and Spain. When differentiated by playing position there is a different RAE incidence in the five championships analyzed

Quantum Iterated Function Systems

Iterated functions system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum iterated functions system (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting a QIFS consists of completely positive maps acting in the space of density operators. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.Comment: 12 pages, 1 figure include

Drip Paintings and Fractal Analysis

It has been claimed [1-6] that fractal analysis can be applied to unambiguously characterize works of art such as the drip paintings of Jackson Pollock. This academic issue has become of more general interest following the recent discovery of a cache of disputed Pollock paintings. We definitively demonstrate here, by analyzing paintings by Pollock and others, that fractal criteria provide no information about artistic authenticity. This work has also led to two new results in fractal analysis of more general scientific significance. First, the composite of two fractals is not generally scale invariant and exhibits complex multifractal scaling in the small distance asymptotic limit. Second the statistics of box-counting and related staircases provide a new way to characterize geometry and distinguish fractals from Euclidean objects

Irreversibility in a simple reversible model

This paper studies a parametrized family of familiar generalized baker maps, viewed as simple models of time-reversible evolution. Mapping the unit square onto itself, the maps are partly contracting and partly expanding, but they preserve the global measure of the definition domain. They possess periodic orbits of any period, and all maps of the set have attractors with well defined structure. The explicit construction of the attractors is described and their structure is studied in detail. There is a precise sense in which one can speak about absolute age of a state, regardless of whether the latter is applied to a single point, a set of points, or a distribution function. One can then view the whole trajectory as a set of past, present and future states. This viewpoint is then applied to show that it is impossible to define a priori states with very large "negative age". Such states can be defined only a posteriori. This gives precise sense to irreversibility -- or the "arrow of time" -- in these time-reversible maps, and is suggested as an explanation of the second law of thermodynamics also for some realistic physical systems.Comment: 15 pages, 12 Postscript figure