Abstract
In this paper we review several contributions made in the field of discrete dynamical systems, inspired by harmonic analysis. Within discrete dynamical systems, we focus exclusively on quadratic maps, both one-dimensional (1D) and two-dimensional (2D), since these maps are the most widely used by experimental scientists. We first review the applications in 1D quadratic maps, in particular the harmonics and antiharmonics introduced by Metropolis, Stein and Stein (MSS). The MSS harmonics of a periodic orbit calculate the symbolic sequences of the period doubling cascade of the orbit. Based on MSS harmonics, Pastor, Romera and Montoya (PRM) introduced the PRM harmonics, which allow to calculate the structure of a 1D quadratic map. Likewise, we review the applications in 2D quadratic maps. In this case both MSS harmonics and PRM harmonics deal with external arguments instead of with symbolic sequences. Finally, we review pseudoharmonics and pseudoantiharmonics, which enable new interesting applications.
Highlights
In this paper we review a branch of harmonic analysis applied to discrete dynamical systems
We focus exclusively on quadratic maps, both one-dimensional (1D) and two-dimensional (2D), since these maps are the most widely used by experimental scientists
We review a branch of the harmonic analysis applied to dynamical systems
Summary
In this paper we review a branch of harmonic analysis applied to discrete dynamical systems. To identify a HC (or a Misiurewicz point) in a 2D quadratic map we normally use the external arguments (EAs) associated to the external rays of Douady and Hubbard [15,20,21] that land in the cusp/root points of the cardioids/discs (or in the Misiurewicz points) These EAs are given as rational numbers with odd denominator in the case of hyperbolic components, and with even denominator in the case of Misiurewicz points. While MSS harmonics were introduced by using the logistic map, PRM harmonics were introduced by using the real Mandelbrot map These PRM harmonics and antiharmonics are a powerful tool that can help us in both the ordering of the periodic orbits of the chaotic region (and those of the periodic region as in the case of the MSS harmonics) and the calculation of symbolic sequences of these orbits. These two new tools will allow new orderings and new calculations in this chaotic region
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