Abstract
In this paper we consider the metric entropies of the maps of an iterated function system deduced from a black hole which are known the Bekenstein–Hawking entropies and its subleading corrections. More precisely, we consider the recent model of a Bohr-like black hole that has been recently analysed in some papers in the literature, obtaining the intriguing result that the metric entropies of a black hole are created by the metric entropies of the functions, created by the black hole principal quantum numbers, i.e., by the black hole quantum levels. We present a new type of topological entropy for general iterated function systems based on a new kind of the inverse of covers. Then the notion of metric entropy for an Iterated Function System () is considered, and we prove that these definitions for topological entropy of IFS’s are equivalent. It is shown that this kind of topological entropy keeps some properties which are hold by the classic definition of topological entropy for a continuous map. We also consider average entropy as another type of topological entropy for an which is based on the topological entropies of its elements and it is also an invariant object under topological conjugacy. The relation between Axiom A and the average entropy is investigated.
Highlights
This article begins with the quantum black hole (BH) physics
Model [1,2,3], we see that the Bekenstein–Hawking entropy and its subleading corrections is a metric entropy of an iterated function system, and we see that the metric entropy of a BH is function of the BH principal quantum number
In the present paper we extend the notion of topological entropy to a finite set of continuous functions on X which is called an Iterated Function System (IFS) [9,10]
Summary
This article begins with the quantum black hole (BH) physics. Referring to the recent Bohr-like BH model [1,2,3], we see that the Bekenstein–Hawking entropy and its subleading corrections is a metric entropy of an iterated function system, and we see that the metric entropy of a BH is function of the BH principal quantum number (the “overtone” number). Topological entropy for a continuous map f : X → X on a compact metric space ( X, d) has been considered from different viewpoints [4,5,6,7,8]. The intuitive picture is more than a picture as QNMs can be really interpreted in terms of BH quantum levels in a BH model somewhat similar to the semi-classical Bohr model of the structure of a hydrogen atom [1,2,3]. = 1), for large values of the principal quantum number n (i.e., for excited BHs), the energy levels of the Schwarzschild BH which is interpreted as gravitational hydrogen atom are given by [1,2,3].
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