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Original Articles

Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family

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References

  • [Arbieto and Matheus 04] A. Arbieto and C. Matheus. “Decidability of Chaos for Some Families of Dynamical Systems.” Found. Comput. Math. 4: 3 (2004), 269–275.
  • [Benedicks and Carleson 85] M. Benedicks and L. Carleson. “On Iterations of 1 −ax2 on ( − 1, 1).” Ann. Math. 122 (1985), 1–25.
  • [Day et al. 08] S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka, and P. Pilarczyk. “Quantitative Hyperbolicity Estimates in One-Dimensional Dynamics.” Nonlinearity 21 (2008), 1967–1987.
  • [Graczyk and Swiatek 97] J. Graczyk and G. Swiatek. “Generic Hyperbolicity in the Logistic Family.” Ann. Math. 146 (1997), 1–52.
  • [Huang 11] Y.-R. Huang. “Measure of Parameters with Acim Nonadjacent to the Chebyshev Value in the Quadratic Family.” PhD thesis, University of Maryland, 2011.
  • [Jakobson 81] M. V. Jakobson. “Absolutely Continuous Invariant Measures for One-Parameter Families of One-Dimensional Maps.” Commun. Math. Phys. 81 (1981), 39–88.
  • [Jakobson 01] M. V. Jakobson. “Piecewise Smooth Maps with Absolutely Continuous Invariant Measures and Uniformly Scaled Markov Partitions.” Proc. Symp. Pure Math. 69 (2001), 825–881.
  • [Jakobson 04] M. V. Jakobson. “Parameter Choice for Families of Maps with Many Critical Points.” In Modern Dynamical Systems and Applications. Cambridge: Cambridge University Press, 2004.
  • [Karp 78] R. M. Karp. “A Characterization of the Minimum Cycle Mean in a Digraph.” Discrete Math. 23: 3 (1978), 309–311.
  • [Kozlovski 03] O. Kozlovski. “Axiom A Maps Are Dense in the Space of Unimodal Maps in the Ck Topology.” Ann. Math. 157 (2003), 1–43.
  • [Kozlovski 07] O. Kozlovski, W. Shen, and S. van Strien. “Density of Hyperbolicity in Dimension One.” Ann. Math. 166 (2007), 145–182.
  • [Luzzatto and Tucker 99] S. Luzzatto and W. Tucker. “Non-Uniformly Expanding Dynamics in Maps with Singularities and Criticalities.” Institut Des Hautes Etudes Scientifiques. Publications Mathématiques 89 (1999), 179–226. 2000.
  • [Luzzatto and Viana 00] S. Luzzatto and M. Viana. “Positive Lyapunov Exponents for Lorenz-Like Families with Criticalities.” Astérisque 261: xiii (2000), 201–237.
  • [Luzzatto and Takahashi 06] S. Luzzatto and H. Takahashi. “Computable Conditions for the Occurrence of Non-Uniform Hyperbolicity in Families of One-Dimensional Maps.” Nonlinearity 19 (2006), 1657–1695.
  • [Lyubich 97a] M. Lyubich. “Dynamics of Quadratic Polynomials. I.” Acta Math. 178 (1997), 185–247.
  • [Lyubich 97b] M. Lyubich. “Dynamics of Quadratic Polynomials. II.” Acta Math. 178 (1997), 247–297.
  • [Lyubich 02] M. Lyubich. “Almost Every Real Quadratic Map is Either Regular or Stochastic.” Ann. Math. 156: 1 (2002), 1–78.
  • [Mañé 85] R. Mañé. “Hyperbolicity, Sinks and Measure in One-Dimensional Dynamics.” Commun. Math. Phys. 100: 4 (1985), 495–524.
  • [Nowicki and Sands 98] T. Nowicki and D. Sands. “Non-Uniform Hyperbolicity and Universal Bounds for S-Unimodal Maps.” Inventiones Mathematicae 132: 3 (1998), 633–680.
  • [Pacifico et al. 98] M. J. Pacifico, A. Rovella, and M. Viana. “Infinite-Modal Maps with Global Chaotic Behavior.” Ann. Math. Second Ser. 148:2 (1998), 441–484.
  • [Pilarczyk 10] P. Pilarczyk. “Parallelization Method for a Continuous Property.” Found. Comput. Math. 10: 1 (2010), 93–114.
  • [Pilarczyk] P. Pilarczyk. “A Space-Efficient Algorithm for Computing the Minimum Cycle Mean in a Directed Graph.” Submitted.
  • [Rovella 93] A. Rovella. “The Dynamics of Perturbations of the Contracting Lorenz Attractor.” Boletim Da Sociedade Brasileira De Matemática. Nova Série 24: 2 (1993), 233–259.
  • [Rychlik 88] M. R. Rychlik. “Another Proof of Jakobson’s Theorem and Related Results.” Ergod. Theory Dyn. Syst. 8 (1988), 93–109.
  • [Simó and Tatjer 91] C. Simó and J. C. Tatjer. “Win-dows of Attraction of the Logistic Map.” Paper presented at the European Conference on Iteration Theory, Batschuns, 335–342, 1989. World Sci. Publ., River Edge, NJ.
  • [Shishikura 12] M. Shishikura. “A Proof of Jakobson’s Theorem via Yoccoz Puzzles and the Measure of Stochastic Parameters.” Conference slides. 2012.
  • [Thunberg 99] H. Thunberg. Positive Exponent in Families with Flat Critical Point. Ergod. Theory Dyn. Syst. 19 (1999), 767–807.
  • [Tsujii 93] M. Tsujii. “Positive Lyapunov Exponents in Families of One-Dimensional Dynamical Systems.” Invent. Math. 111 (1993), 113–137.
  • [Tucker and Wilczak 09] W. Tucker and D. Wilczak.” A Rigorous Lower Bound for the Stability Regions of the Quadratic Map.” Physica D 238: 18 (2009), 1923–1936.
  • [Ulam and von Neumann 47] S. Ulam and J. von Neumann. “On Combination of Stochastic and Deterministic Processes.” Bull AMS 53: 11 (1947), 1120.

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