In these notes we first offer an overview of two core areas in the dynamics of probability measure preserving systems, the Kolmogorov-Sinai theory of entropy and the theory of orbit equivalence. Entropy is a nontrivial invariant that, said simply, measures the exponential growth rate of the number of orbits in a dynamical system, a very rough measure of the complexity of the orbit structure. On...

We review recent results on the relationships between the microscopic chaos in the motion of atoms or molecules in fluids and the transport properties sustained across these macroscopic systems. In the escape-rate formalism, the transport coefficients can be expressed in terms of the positive Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time or the Hausdorff dimension of a fractal ...

The hallmark of deterministic chaos is that it creates information—the rate being given by the Kolmogorov-Sinai metric entropy. Since its introduction half a century ago, the metric entropy has been used as a unitary quantity to measure a system’s intrinsic unpredictability. Here, we show that it naturally decomposes into two structurally meaningful components: A portion of the created informat...

Abstract. Efficient hierarchical architectures for reconfigurable and adaptive multi-agent networks require dynamic cluster formation among the set of nodes (agents). In the absence of centralised controllers, this process can be described as self-organisation of dynamic hierarchies, with multiple cluster-heads emerging as a result of inter-agent communications. Decentralised clustering algorit...

On an influential account, chaos is explained in terms of random behaviour; and random behaviour in turn is explained in terms of having positive Kolmogorov-Sinai entropy (KSE). Though intuitively plausible, the association of the KSE with random behaviour needs justification since the definition of the KSE does not make reference to any notion that is connected to randomness. I provide this ju...

Statistical entropy was introduced by Shannon as a basic concept in information theory, measuring the average missing information on a random source. Extended into an entropy rate, it gives bounds in coding and compression theorems. I here present how statistical entropy and entropy rate relate to other notions of entropy, relevant either to probability theory (entropy of a discrete probability...

In this paper, we investigate topological complexity of saturated set from the viewpoints upper capacity entropy and packing entropy. We obtain that if a dynamical system $ (X, T) satisfies almost product property, then for each nonempty compact convex subset K invariant measures, following two formulas hold G_K $:$ \begin{equation*} h_{top}^{UC}(T, G_{K}) = h_{top}(T, X)\ \mathrm{and}\ h_{top}...

Abstract The Lyapunov exponent is used to quantify the chaos of a dynamical system, by characterizing exponential sensitivity an initial point on system. However, we cannot directly compute for system without its equation, although some estimation methods do exist. Information dynamics introduces entropic degree measure strength can be with practical time series. It may seem like kind finite sp...

We adapt the Kolmogorov-Sinai entropy to the non-extensive perspective recently advocated by Tsallis. The resulting expression is an average on the invariant distribution, which should be used to detect the genuine entropic index Q. We argue that the condition Q > 1 is time dependent. 05.20.-y,05.45.-a,05.60.Cd Typeset using REVTEX 1 The importance of establishing a connection between dynamics ...