Anosov actions on noncommutative algebras

Abstract

An axiomatic framework for a quantum mechanical extension to the theory of Anosov systems is constructed, and it is shown that this retains some of the characteristic features of its classical counterpart, e.g., nonvanishing Lyapunov exponents, a vectorial K-property, and exponential clustering. The effects of quantization are investigated on two prototype examples of Anosov systems, namely, the iterations of an automorphism of the torus (the ‘‘Arnold Cat’’ model) and the free dynamics of a particle on a surface of negative curvature. It emerges that the Anosov property survives quantization in the case of the former model, but not of the latter one. Finally, we show that the modular dynamics of a relativistic quantum field on the Rindler wedge of Minkowski space is that of an Anosov system.

On the computation of quantum characteristic exponents

“…Other definitions of characteristic exponents in infinite-dimensional spaces have been proposed by several authors [8] [9] [10] [11] [12] [13]. They characterize several aspects of the dynamics of linear and non-linear systems.…”

Semiclassical properties and chaos degree for the quantum Baker’s map

“…We will use the simpler definition which in the position representation is q j |ψ 0 = C exp −q 2 j /2 (compare with (9)). Here C is a normalization constant.…”

“…The quantum-classical correspondence for dynamical systems has been studied for many years, see for example [5,6,7,8,9,10] and reference therein. A significant progress in understanding of this correspondence has been achieved in the WKB approach when one considers the Planck constant h as a small variable parameter.…”

Quantum geometry of noncommutative Bernoulli shifts

“…However, in a general situation there exist no canonical derivations which could span a noncommutative analog of the tangent space. The nice special case is the noncommutative torus given by the irrational rotation algebra A θ generated by two unitaries and possessing a corresponding two dimensional "tangent space" of outer derivations [8]. In the example discussed below such a natural system of derivations has not been found and instead we introduce a weaker geometrical structure supporting the notion of Lyapunov exponents.…”

“…Quantum Lyapunov exponents. There were several attempts to define quantum Lyapunov exponents [8,11]. In the abstract context of a C * -algebraic dynamical system (A, Θ) with the additional structure of unbounded derivations {D i } over a Θ-invariant smooth subalgebra A ∞ ⊂ A the Lyapunov exponents can be defined as [8] …”