In the paper we define three kinds of entropy of a fuzzy dynamical system using different entropies of fuzzy partitions. It is shown that different definitions of the entropy of fuzzy partitions lead to different notions of entropies of fuzzy dynamical systems. The relationships between these entropies are studied and connections with the classical case are mentioned as well. Finally, an analog...

The development of quantum information theory motivated the extension to the quantum realm of notions from algorithmic complexity theory. Because of the structure of quantum mechanics, several inequivalent generalizations are possible. The same phenomenon characterizes the extension of the dynamical entropy of Kolmogorov and Sinai. In the following, we shall examine the relations between the qu...

Slow (logarithmic) relaxation from a highly excited state is studied in a Hamiltonian system with many degrees of freedom. The relaxation time is shown to increase as the exponential of the square root of the energy of excitation, in agreement with the Boltzmann-Jeans conjecture, while it is found to be inversely proportional to residual Kolmogorov-Sinai entropy, introduced in this Rapid Commun...

1 Introduction Kolmogorov-Sinai entropy (K-S entropy) is used to analyze radio wave intensity time series of quasars. K-S entropy measures the rate of change of probability with the change of dimension that trajectory points in the embedding space stay within a distance r [1]. The K-S entropy analysis method is a powerful formalism for describing dynamics of time series [1]. m-dimensional K-S e...

We give the relationship between symbolic dynamics and chaos synchronization. If the capacity of the channel which onedirectionally connects two chaotic systems with the aim of synchronizing them is larger than Kolmogorov–Sinai entropy of the driving system, then the synchronization error can be made arbitrarily small.

The von Neumann entropy production for a quantum mechanical kicked rotor coupled to a thermal environment is calculated. This rate of entropy increase is shown to be a good criterion to distinguish between quantum mechanical counterparts of chaotic and regular classical motion. We show that for high temperatures the entropy production rate increases linearly with the Kolmogorov-Sinai entropy of...

Several quantum dynamical entropies have been proposed that extend the classical Kolmogorov–Sinai (dynamical) entropy. The same scenario appears in relation to the extension of algorithmic complexity theory to the quantum realm. A theorem of Brudno establishes that the complexity per unit time step along typical trajectories of a classical ergodic system equals the KS-entropy. In the following,...

We review known results and derive some new ones about the mean free path, Kolmogorov-Sinai entropy and Lyapunov exponents for billiard type dynamical systems. We focus on exact and asymptotic formulas for these quantities. The dynamical systems covered in the paper include periodic Lorentz gas, stadium and its modifications, and the gas of hard balls. Some open questions and numerical observat...

We give a generator-free formulation of sofic measure entropy using finite partitions and establish a Kolmogorov-Sinai theorem. We also show how to compute the values for general Bernoulli actions in a concise way using the arguments of Bowen in the finite base case.