In this paper we introduce a new functional invariant of discrete time dynamical systems — the so-called t-entropy. The main result is that this t-entropy is the Legendre dual functional to the logarithm of the spectral radius of the weighted shift operator on L1(X,m) generated by the dynamical system. This result is called the Variational principle and is similar to the classical variational p...

In a previous paper the authors developed an operator-algebraic approach to Lewis Bowen’s sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure-preserving transformations on a standard probability space. We show here that these measure and topological entropy invariants both coincide with their classical...

Recently there has been an extensive study on Relative Semi-Dynamical Systems (RSD-Systems). In this paper, we explore some topological properties of RSD-systems. Here, in particular, minimal RSD-systems are characterized and transitive homeomorphisms are investigated. Moreover, α-level relative topological entropy is extended to RSD-systems. Finally, as a computational example, we develop an R...

A closed expression for the density operator of the damped harmonic oscillator is extracted from the master equation based on the Lindblad theory for open quantum systems. The entropy and effective temperature of the system are subsequently calculated and their temporal behaviour is surveyed by showing how these quantities relax to their equilibrium values. The entropy for a state characterized...

A unitary operator U acting on the weakly compact unit ball B of a Hilbert space H gives a topological dynamical system (B, V). We prove in this note that for every ergodic u-invariant measure m, the dynamical system (B, U, m) is conjugated to a strictly ergodic group translation. Especially, the topological entropy of (B, U) is zero. As an application, it follows that any flow on the unit ball...

We analyze entropy solutions for a class of Lévy mixed hyperbolicparabolic equations containing a non-local (or fractional) diffusion operator originating from a pure jump Lévy process. For these solutions we establish uniqueness (L1 contraction property) and continuous dependence results.

For timed languages, we define size measures: volume for languages with a fixed finite number of events, and entropy (growth rate) as asymptotic measure for an unbounded number of events. These measures can be used for quantitative comparison of languages, and the entropy can be viewed as information contents of a timed language. For languages accepted by deterministic timed automata, we give e...

We consider the following question: what is the least biased stochastic process over the runs of a given timed automaton? By “least biased”, we mean the one that generates runs with a distribution which is closest to uniform (w.r.t. volume). To answer this question, we first define stochastic processes over runs of timed automata and their entropy, the definition of which is inspired by Shannon...