By Ioannis Antoniou, Baidyanath Misra, Zdzislaw Suchanecki

The time operator is a robust technique for the research of risky dynamics and chaos idea. This synthesis of classical and glossy fabric offers a hugely utilized method of the improvement of either spectral research and prediction for chaotic complicated structures for which the time operator is defined by way of a semigroup of operators.

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Let us, however, add that a great variety of systems of physical interest, including infinite ideal gas, hard rods system, a hard sphere gas and geodesic flow on a compact, connected smooth Riemanian manifold with negative curvature are known to be K-flows. Geodesic flows have been studied since the beginning of the 20th century, starting with Hadamard. In the case the manifold X has dim X = 2, the geodesic flow has an especially simple interpretation. It describes the notion of a point that moves on the surface X in the absence of external forces and without friction.

Since the entropy is a quantitative measure of information carried by a σ-algebra we can say that the filtration is innovative if H(Fs ) < H(Ft ) , for each s < t . In this way, by introducing entropy, we give a rigorous meaning to innovation in each system having countably generated filtration. However, from the practical point of view entropy, as the criterion of innovations, is not so useful as it seems to be. This concerns especially stochastic processes. Although each stochastic process determines a filtration it does not mean that the σ-algebras are generated by countable partitions.

About the strong convergence of truncated operators). 7). e. 7) we have T ρt |ρt = T ρ|ρ + t . The above equality means that the average age of the evolved state keeps step with the external clock time t. The above theorem suggests the following generalization of time operator. Let H be a (separable) Hilbert space and {Ut } a unitary group of evolution. : Ut D(T ) ⊂ D(T ) and T Ut = Ut T + tUt , for each t. This generalization of the notion of time operator can go further. The group {Ut } can be replaced by a semigroup, and/or the Hilbert space can be replaced by a Banach space.