We study totally ergodic quantum dynamical systems with quasi–discrete spectrum. We investigate the classification problem for such systems in terms of algebraic invariants. The results are noncommutative analogs of (a part of) the theory of Abramov. Supported in part by the National Science Foundation under grant DMS–9801612

If values that we monitor changes during discrete periods (for example, in discrete time intervals), the formula above leads to a difference equation or a dynamical system. In this case, we are dealing with a function that depends on discrete integer values – a sequence. Recall that a sequence of real numbers (indexed by nonnegative integers) can be given by listing its terms (for example, 1, 1...

It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco–Tu and Riemann–Burgers dynamical systems.

In this paper variation of constants formula for the inhomogeneous fuzzy matrix difference system X(n + 1) = A(n)X(n) + F (n) is constructed by considering corresponding matrix difference inclusion Xβ(n+ 1) ∈ AβXβ(n) +Fβ(n), xβ(0) ∈ X0 β after presenting a procedure for solving the fuzzy difference equation by an example. Mathematics Subject Classification: 26E50, 34A12, 34A30

We define and study topological pressure for the non-autonomous discrete dynamical systems given by a sequence {fi} ∞ i=1 of continuous self-maps of a compact metric space. In this paper, we obtain the basic properties and the invariant with respect to equiconjugacy of topological pressure for the nonautonomous discrete dynamical systems.

We give a upper bound of Lebesgue measure V (S(f, h,Ω)) of the set S(f, h,Ω) of points q ∈ Qdh for which the triple (h, q,Ω) is dynamically robust when f is monotonic and satisfies certain condition on some compact subset Ω ∈ R.

A discrete dynamical system in Euclideanm-space generated by the iterates of an asymptotically zero map f , satisfying |f(x)| → 0 as |x| → ∞, must have a compact global attracting set A. The question of what additional hypotheses are sufficient to guarantee that A has a minimal (invariant) subset A that is a chaotic strange attractor is answered in detail for a few types of asymptotically zero ...

The fundamental laws of physics are described from the modern point of view by gauge theories with continuous gauge groups. With some modification the gauge principle can (and should) be applied to discrete dynamical systems with nontrivial symmetries also. We consider specifics of introduction of gauge invariance into W -equivariant discrete dynamical systems with finite sets of states Σ, wher...

The deenitions of isolating block, index pair, and the Conley index, together with the proof of homotopy and additivity properties of the index are generalized for discrete multivalued dynamical systems. That generalization provides a theoretical background of numerical computation used by Mischaikow and Mrozek in their computer assisted proof of chaos in the Lorenz equations, where nitely repr...