Table of contents

The authors define a new statistic, the parametric number variance, which measures the correlation of fluctuations in energy levels as a parameter external to the system is varied. A semiclassical formula is obtained and regimes of universal and system dependent behaviour are predicted. Numerical calculations of the PNV in two model systems are found to be in good agreement with the semiclassical theory.

Let f be a C²-diffeomorphism of a compact surface M, Lambda be a nontrivial locally maximal hyperbolic set of f such that f mod Lambda is topologically transitive, and mu be the equilibrium state corresponding to some Holder continuous function psi on Lambda . Under these conditions the author proves that generically the lower and upper pointwise dimensions of the measure mu , d_mu (x) and d_mu (x), are different (d_mu (x)<d_mu (x)) when x runs over a set which is dense in Lambda and has positive Hausdorff dimension. This fact means, in particular, that one cannot replace the condition 'for mu -almost every x' in Young's theorem by the condition 'for all x in supp mu ' and even 'up to the set of zero Hausdorff dimension'.

The method for computing the topological entropy of the dynamical systems generated by one-dimensional piecewise-continuous piecewise-monotonous maps of interval is proposed. The method is based on the kneading theory and allows one to reduce the computational process to that of seeking the minimum positive zero of some polynomial. A mathematical model of a clock has been studied, which confirmed the high efficiency of the method. The topological entropy is related to the Lyapunov characteristic exponent and to the structure of attractors. The problems involved in the numerical realization of the proposed algorithm are discussed.

A simple example of coupled map lattices generated by expanding maps of the unit interval with some kind of diffusion coupling is considered. It has been stated that this system has an unique invariant mixing measure with absolutely continuous finite-dimensional projections. Here the author proves that probability measures from some natural class weakly converge to this measure under the actions of dynamics. The main idea of the proof is the symbolic representation of his system by two-dimensional lattice model of statistical mechanics. It provides the possibility of applying results from random field theory.

The results of theoretical and experimental investigation of nonlinear effects revealed in solid semiconductor solutions under optical orientation of charge carriers are presented. Those effects caused by hyperfine interaction between the carriers and the crystal lattice nuclei manifest themselves in the appearance of bistability and auto-oscillations of luminescence polarization excited by circularly polarized light. Theoretical analysis of data obtained in the experiments carried out in a strong magnetic field (B approximately 100-1000 G) allowed the authors to determine the values of phenomenological parameters describing the Overhauser effective nucleus field.

The authors study the correlations in the quasi-energy (QE) spectra of systems with dynamical localization, using the quantum kicked rotor (QKR) as a paradigm. The specific spatial structure of the QE eigenstates is taken into account by investigating the local spectrum, which gives each eigenstate an individual weight according to its overlap with some reference state. Two-point correlations in the local spectrum are related by Fourier transform to the time evolution of the probability to stay at the initial state. They devise a scaling theory for this dynamical quantity in the case of the QKR, containing the participation ratio as a single parameter. It implies that the local spectrum is characterized by positive correlations, in contrast to the unbiased spectra in classically chaotic systems with a bounded phase space. This is consistent with recent results on spectral properties of systems with Anderson localization. A scheme for experimental measurements of spectral two-point correlation functions is proposed.

For pt.I see ibid., vol.4, p.59-84 (1991). The authors study correlations in the quasi-energy spectrum of the quantum kicked rotor restricted to a Hilbert space of finite dimension. The spectral correlations depend on the ratio gamma of the localization length to the basis size. They derive semiclassical expressions for the two-point cluster function which interpolate between COE behaviour for gamma to infinity and Poissonian (lack of correlations) for gamma to 0. They show how the diffusive nature of the classical dynamics finds its expression in the quantal spectral correlations.

Let T be a rational map of degree d>or=2 of the Riemann sphere C=C union ( infinity ). The authors develop the theory of equilibrium states for the class of Holder continuous functions f for which the pressure is larger than sup f. They show that there exist a unique conformal measure (reference measure) and a unique equilibrium state, which is equivalent to the conformal measure with a positive continuous density. The associated Perron-Frobenius operator acting on the space of continuous functions is almost periodic and they show that the system is exact with respect to the equilibrium measure.

The authors present a connection between the concepts of determining nodes and inertial manifolds with that of finite difference and finite volumes approximations to dissipative partial differential equations. In order to illustrate this connection they consider the 1D Kuramoto-Sivashinsky equation as a instructive paradigm. They remark that the results presented here apply to many other equations such as the 1D complex Ginzburg-Landau equation, the Chafee-Infante equation, etc.

A variational principle is presented, characterizing odd-dimensional submanifolds of an energy surface invariant under a Hamiltonian flow.