Uniform expansivity outside a critical neighborhood in the quadratic family

article Uniform expansivity outside a critical neighborhood in the quadratic family published yes Ali Golmakani author Stefano Luzzatto author Pawel Pilarczyk author 3768D56A-F248-11E8-B48F-1D18A9856A87 HeEd department Persistent Homology - Images, Data and Maps project 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/. Taylor & Francis2016 eng Experimental Mathematics 1504.00116 00037249050000210.1080/10586458.2015.1048011 252116 - 124 A. Golmakani, S. Luzzatto, P. Pilarczyk, Experimental Mathematics 25 (2016) 116–124. Golmakani A, Luzzatto S, Pilarczyk P. 2016. Uniform expansivity outside a critical neighborhood in the quadratic family. Experimental Mathematics. 25(2), 116–124. Golmakani, A., Luzzatto, S., & Pilarczyk, P. (2016). Uniform expansivity outside a critical neighborhood in the quadratic family. Experimental Mathematics. Taylor & Francis. <a href="https://doi.org/10.1080/10586458.2015.1048011">https://doi.org/10.1080/10586458.2015.1048011</a> A. Golmakani, S. Luzzatto, and P. Pilarczyk, “Uniform expansivity outside a critical neighborhood in the quadratic family,” Experimental Mathematics, vol. 25, no. 2. Taylor & Francis, pp. 116–124, 2016. Golmakani, Ali, et al. “Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family.” Experimental Mathematics, vol. 25, no. 2, Taylor & Francis, 2016, pp. 116–24, doi:<a href="https://doi.org/10.1080/10586458.2015.1048011">10.1080/10586458.2015.1048011</a>. Golmakani A, Luzzatto S, Pilarczyk P. Uniform expansivity outside a critical neighborhood in the quadratic family. Experimental Mathematics. 2016;25(2):116-124. doi:<a href="https://doi.org/10.1080/10586458.2015.1048011">10.1080/10586458.2015.1048011</a> Golmakani, Ali, Stefano Luzzatto, and Pawel Pilarczyk. “Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family.” Experimental Mathematics. Taylor & Francis, 2016. <a href="https://doi.org/10.1080/10586458.2015.1048011">https://doi.org/10.1080/10586458.2015.1048011</a>. 12542018-12-11T11:50:58Z2026-07-07T05:34:16Z