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Template Iterations of Quadratic Maps and Hybrid Mandelbrot Sets

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As a particular problem within the field of non-autonomous discrete systems, we consider iterations of two quadratic maps \(f_{c_0}=z^2+c_0\) and \(f_{c_1}=z^2+c_1\), according to a prescribed binary sequence, which we call a template. We study the asymptotic behavior of the critical orbits and define the Mandelbrot set in this case as the locus for which these orbits are bounded. However, unlike in the case of single maps, this concept can be understood in several ways. For a fixed template, one may consider this locus as a subset of the parameter space in \((c_0,c_1) \in \mathbb {C}^2\); for fixed quadratic parameters, one may consider the set of templates which produce a bounded critical orbit. In this paper, we consider both situations and hybrid combinations of them, and we study basic topological properties of these sets and interpret them in light of potential applications.

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Acknowledgements

The work on this project was supported by SUNY New Paltz, via the Research Scholarship and Creative Activities Program (Grant No. SURE 2016) and the Research and Creative Projects Awards Program, as well as by the Simons Foundation Collaborative Grants for Mathematicians (Grant No. 523763).

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  1. Department of Mathematics, State University of New York at New Paltz, New Paltz, NY, 12561, USA

    Anca Rǎdulescu, Kelsey Butera & Brandee Williams

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  1. Anca Rǎdulescu
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  2. Kelsey Butera
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  3. Brandee Williams
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Correspondence to Anca Rǎdulescu.

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Communicated by Amy Radunskaya.

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Rǎdulescu, A., Butera, K. & Williams, B. Template Iterations of Quadratic Maps and Hybrid Mandelbrot Sets. J Nonlinear Sci 31, 22 (2021). https://doi.org/10.1007/s00332-020-09675-0

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