Extensions of the Shannon Entropy and the Chaos Game Algorithm to Hyperbolic Numbers Plane

TL;DR

This paper extends the classical Chaos game algorithm and Shannon entropy to the hyperbolic numbers plane, exploring hyperbolic-valued probabilities and entropy interpretations.

Contribution

It introduces hyperbolic extensions of key concepts in probability, entropy, and the Chaos game algorithm, expanding their applicability to hyperbolic number systems.

Findings

Hyperbolic-valued probabilities are defined and interpreted.

Extensions of Shannon entropy to hyperbolic numbers are developed.

The hyperbolic Chaos game algorithm is formulated and analyzed.

Abstract

In this paper we provide extensions to hyperbolic numbers plane of the classical Chaos game algorithm and the Shannon entropy. Both notions connected with that of probability with values in hyperbolic number, introduced by D. Alpay et al \cite{eluna}. Within this context, particular attention has been paid to the interpretation of the hyperbolic valued probabilities and the hyperbolic extension of entropy as well.