Rényi entropies of aperiodic dynamical systems
This paper considers generalized entropies of dynamical systems introduced by Rényi, using the one-parameter family of generalized entropy functions defined by: \[ H_q(p_1,\dots,p_n)=-\frac1{q-1}\log(\sum_{i=1}^n p_i^1),\qquad q\neq 1. \] The usual Shannon entropy function is obtained as a limit of these functions: \(\lim_{q\rightarrow 1}H_q(p_1,\dots,p_n)\). This paper is one in a series which attempts to define new metric invariants for dynamical systems by introducing a generalized entropy of a dynamical system using the Rényi entropy functions. In an earlier paper [Nonlinearity 11, 771--782 (1998; Zbl 0943.37004)], the authors showed that simply replacing \(H_1\) with \(H_q\) yields no new results for ergodic transformations of positive entropy. For \(q<1\) one simply obtains \(+\infty\) and for \(q\geq 1\) one recovers the usual measure theoretic entropy. In this paper, they extend this result to nonergodic transformations where the situation is slightly different. It turns out that for \(q>1\) the generalized Rényi entropy can, potentially, detect ergodicity. Namely, if the hypothesis of ergodicity is weakened to aperiodicity, then for \(q<1\) one obtains \(+\infty\), for \(q=1\) one recovers the usual measure theoretic entropy, but for \(q>1\) the generalized entropy yields the essential infimum of the usual entropies of the ergodic components of \(T\). Since the usual measure-theoretic entropy is the average value of the entropies of the ergodic components, this result allows for the existence of examples of nonergodic transformations \(T\) where the generalized entropy is strictly smaller than the measure theoretic entropy of \(T\). Unfortunately, the authors do not have an example for which this observation is (in their words) useful. The paper is well written, the techniques are interesting, and the result is surprising and interesting in the sense that it suggests the existence of an entropy-based invariant which can detect ergodicity.
scientific article; zbMATH DE number 1786383
scientific article