Uniform expansion estimates in the quadratic map with the smallest critical neighborhood for which the expansion exponent λ0 is greater than 0.1

This dataset contains selected results of numerical computations described in the paper "Quantitative hyperbolicity estimates in one-dimensional dynamics" by S. Day, H. Kokubu, S. Luzzatto, K. Mischaikow, H. Oka, P. Pilarczyk, published in Nonlinearity, Vol. 21, No. 9 (2008), 1967-1987, doi: 10.1088/0951-7715/21/9/002. Using an approach based on graph algorithms described in the paper, comprehensive computation was conducted to obtain a rigorous lower bound on the expansion exponent λ (Statement 1 in the paper), the constant C (Statement 1 in the paper), and the adjusted exponent λ0 (Statement 2 in the paper). The size of the partition into which the domain of the map was subdivided is denoted by K. The radius of the critical neighborhood is denoted by δ. The key parameter of the map is denoted by a. The parallelization framework introduced in the paper "Parallelization method for a continuous property" by P. Pilarczyk, as published in Foundations of Computational Mathematics, Vol. 10, No. 1 (2010), 93–114, doi: 10.1007/s10208-009-9050-8, was used in order to use several CPUs at a time. Note that the data records are not sorted. In this specific computation, the quadratic map was analyzed with K=5000, a∈[1.7,2.0], and δ chosen to be as small as possible, assuming that still λ0>0.1 can be obtained. The data is provided in plain text format. Each line with comments begins with a semicolon, each line with a single data record begins with an asterisk. Each data record consists of the following items, separated with the space: the asterisk that indicates the beginning of a data record the identifier of the data record in the format level:number (e.g., 4:12) the left endpoint of the parameter interval (minimal parameter value) the right endpoint of the parameter interval (maximal parameter value) the total number of intervals that cover both the critical neighborhood and the remainder of the domain of the map the diameter δ of the critical neighborhood the computed expansion exponent λ the computed value of log C (0 if not computed) the computed value of λ0 (0 if not computed) the computation time (in seconds) The actual software that was used to obtain the results and also some illustrations are available at http://www.pawelpilarczyk.com/unifexp/.

本数据集包含S. Day、H. Kokubu、S. Luzzatto、K. Mischaikow、H. Oka与P. Pilarczyk发表于《非线性（Nonlinearity）》第21卷第9期（2008年），页码1967-1987，DOI为10.1088/0951-7715/21/9/002的论文《一维动力学中的定量双曲性估计（Quantitative hyperbolicity estimates in one-dimensional dynamics）》中所述数值计算的精选结果。本研究采用该论文中介绍的基于图算法的方法，开展了全面计算，以获得展开指数（expansion exponent）λ、常数C以及调整后指数（adjusted exponent）λ₀的严谨下界，对应论文中的陈述1与陈述2。映射的定义域被划分的分区大小记为K，临界邻域的半径记为δ，该映射的关键参数记为a。本次计算采用了P. Pilarczyk发表于《计算数学基础（Foundations of Computational Mathematics）》第10卷第1期（2010年），页码93-114，DOI为10.1007/s10208-009-9050-8的论文《连续性质的并行化方法（Parallelization method for a continuous property）》中提出的并行化框架，以实现多CPU并行计算。请注意，本数据集的数据记录未经过排序。本次具体计算针对二次映射开展分析，其中K=5000，参数a的取值范围为[1.7, 2.0]，且在保证仍可获得λ₀>0.1的前提下，将δ取为最小可能值。本数据集以纯文本格式提供：所有注释行均以分号（;）开头，单条数据记录所在行则以星号（*）开头。每条数据记录由以下以空格分隔的字段组成：标记数据记录起始的星号、格式为level:number（例如4:12）的数据记录标识符、参数区间左端点（最小参数值）、参数区间右端点（最大参数值）、覆盖临界邻域与映射定义域其余部分的总区间数、临界邻域的直径δ、计算得到的展开指数λ、log C的计算值（未计算时为0）、λ₀的计算值（未计算时为0）以及计算耗时（单位：秒）。用于生成该结果的实际软件及部分图示可通过网址http://www.pawelpilarczyk.com/unifexp/获取。