---
_id: '1254'
abstract:
- lang: eng
  text: 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/.
acknowledgement: "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\r\nMathematics
  \ of  Kyoto  University  for  providing  access\r\nto  their  server  for  conducting
  \ computations  for  this\r\nproject."
article_processing_charge: No
arxiv: 1
author:
- first_name: Ali
  full_name: Golmakani, Ali
  last_name: Golmakani
- first_name: Stefano
  full_name: Luzzatto, Stefano
  last_name: Luzzatto
- first_name: Pawel
  full_name: Pilarczyk, Pawel
  id: 3768D56A-F248-11E8-B48F-1D18A9856A87
  last_name: Pilarczyk
citation:
  ama: Golmakani A, Luzzatto S, Pilarczyk P. Uniform expansivity outside a critical
    neighborhood in the quadratic family. <i>Experimental Mathematics</i>. 2016;25(2):116-124.
    doi:<a href="https://doi.org/10.1080/10586458.2015.1048011">10.1080/10586458.2015.1048011</a>
  apa: Golmakani, A., Luzzatto, S., &#38; Pilarczyk, P. (2016). Uniform expansivity
    outside a critical neighborhood in the quadratic family. <i>Experimental Mathematics</i>.
    Taylor &#38; Francis. <a href="https://doi.org/10.1080/10586458.2015.1048011">https://doi.org/10.1080/10586458.2015.1048011</a>
  chicago: Golmakani, Ali, Stefano Luzzatto, and Pawel Pilarczyk. “Uniform Expansivity
    Outside a Critical Neighborhood in the Quadratic Family.” <i>Experimental Mathematics</i>.
    Taylor &#38; Francis, 2016. <a href="https://doi.org/10.1080/10586458.2015.1048011">https://doi.org/10.1080/10586458.2015.1048011</a>.
  ieee: A. Golmakani, S. Luzzatto, and P. Pilarczyk, “Uniform expansivity outside
    a critical neighborhood in the quadratic family,” <i>Experimental Mathematics</i>,
    vol. 25, no. 2. Taylor &#38; Francis, pp. 116–124, 2016.
  ista: Golmakani A, Luzzatto S, Pilarczyk P. 2016. Uniform expansivity outside a
    critical neighborhood in the quadratic family. Experimental Mathematics. 25(2),
    116–124.
  mla: Golmakani, Ali, et al. “Uniform Expansivity Outside a Critical Neighborhood
    in the Quadratic Family.” <i>Experimental Mathematics</i>, vol. 25, no. 2, Taylor
    &#38; Francis, 2016, pp. 116–24, doi:<a href="https://doi.org/10.1080/10586458.2015.1048011">10.1080/10586458.2015.1048011</a>.
  short: A. Golmakani, S. Luzzatto, P. Pilarczyk, Experimental Mathematics 25 (2016)
    116–124.
corr_author: '1'
das_tickbox: '1'
date_created: 2018-12-11T11:50:58Z
date_published: 2016-04-02T00:00:00Z
date_updated: 2026-07-07T05:34:16Z
day: '02'
department:
- _id: HeEd
doi: 10.1080/10586458.2015.1048011
ec_funded: 1
external_id:
  arxiv:
  - '1504.00116'
  isi:
  - '000372490500002'
intvolume: '        25'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1504.00116
month: '04'
oa: 1
oa_version: Preprint
page: 116 - 124
project:
- _id: 255F06BE-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '622033'
  name: Persistent Homology - Images, Data and Maps
publication: Experimental Mathematics
publication_status: published
publisher: Taylor & Francis
publist_id: '6071'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Uniform expansivity outside a critical neighborhood in the quadratic family
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2016'
...