Golmakani, Ali; Luzzatto, Stefano; Pilarczyk, PawelISTA
We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We first compute a radius of the critical neighborhood outside which the map is uniformly expanding. This radius is taken as small as possible, yet large enough for our numerical procedure to succeed in proving that the expansivity exponent outside this neighborhood is positive. Then, for each of the intervals, we compute a lower bound for this expansivity exponent, valid for all the parameters in that interval. We illustrate and study the distribution of the radii and the expansivity exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/.
AG and PP were partially supported by Abdus Salam International Centre for Theoretical Physics (ICTP). Additionally, AG was supported by BREUDS, and research conducted by PP has received funding from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE—Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Fundação para a Ciência e a Tecnologia (FCT) in the framework of the research project FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008); and from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 622033. The authors gratefully acknowledge the Department of Mathematics of Kyoto University for providing access to their server for conducting computations for this project.
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Golmakani A, Luzzatto S, Pilarczyk P. Uniform expansivity outside a critical neighborhood in the quadratic family. Experimental Mathematics. 2016;25(2):116-124. doi:10.1080/10586458.2015.1048011
Golmakani, A., Luzzatto, S., & Pilarczyk, P. (2016). Uniform expansivity outside a critical neighborhood in the quadratic family. Experimental Mathematics. Taylor and Francis. https://doi.org/10.1080/10586458.2015.1048011
Golmakani, Ali, Stefano Luzzatto, and Pawel Pilarczyk. “Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family.” Experimental Mathematics. Taylor and Francis, 2016. https://doi.org/10.1080/10586458.2015.1048011.
A. Golmakani, S. Luzzatto, and P. Pilarczyk, “Uniform expansivity outside a critical neighborhood in the quadratic family,” Experimental Mathematics, vol. 25, no. 2. Taylor and Francis, pp. 116–124, 2016.
Golmakani A, Luzzatto S, Pilarczyk P. 2016. Uniform expansivity outside a critical neighborhood in the quadratic family. Experimental Mathematics. 25(2), 116–124.
Golmakani, Ali, et al. “Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family.” Experimental Mathematics, vol. 25, no. 2, Taylor and Francis, 2016, pp. 116–24, doi:10.1080/10586458.2015.1048011.