Harmonic structure of one-dimensional quadratic maps

Operating with external arguments in the Mandelbrot set antenna

“…Indeed, as we saw before, the order i MSS-harmonic corresponds to the i th disk of the period doubling cascade which has a period 2 i p, and therefore a pattern with a length 2 i times the original. However, the Fourier harmonics, except for the first harmonic, are all of them last appearance cardioids in the corresponding chaotic band [23], and the order i F-harmonic has a period (i + 1)p, and therefore has a pattern with a length i + 1 times the original. As we did in the case of the MSS-harmonics, we shall introduce now the F-harmonics of the external arguments of an antenna hyperbolic component.…”

“…In order to calculate symbolic sequences in 1D quadratic maps F-harmonics are a very powerful tool since starting from the main cardioid, of simbolic sequence C, we can generate the successive chaotic bands and calculate the pattern of any of their structural components (last appearance hyperbolic components) [23].…”

“…the family tree. In order to do so, we need to use the composition rules of the symbolic sequences that were introduced by us [23]. In this paper we also introduce new composition rules, now applied to external arguments.…”

“…We have studied one-dimensional (1D) quadratic maps by means of the hyperbolic components [7], Misiurewicz points [20] and other notable points of the Mandelbrot set which intersect with the real axis [21,22,23,24,25,26,27]. Figure 2 shows the neighborhood of this Mandelbrot set subset, that we call "antenna".…”

“…By using this harmonic tool we could calculate the symbolic sequences of all the structural components (and the Misiurewicz points that separates them) of the 1D quadratic maps; i. e., we could determine the structure of such maps [23].…”