Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family

Abstract

Abstract. 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 compute a possibly small radius of the critical neighborhood, and a lower bound for the corresponding expansivity exponent outside this neighborhood, valid for all the parameters in each of the intervals. We illustrate and study the distribution of the radii and these 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/.Key words and phrases: dynamical system, quadratic map family, uniform expansivity.We report on some rigorous numerical results concerning expansivity exponents for various parameter intervals for the dynamics outside a critical neighborhood in the quadratic family of maps. For completeness and in order to motivate our study, we start in Section 1 with some general facts on the quadratic family. The reader who is already familiar with the basic background may wish to skip directly to Section 2.

A space-efficient algorithm for computing the minimum cycle mean in a directed graph

“…In order to confirm the usefulness of Algorithm 3.1 and to compare its demand for resources with Karp's original algorithm [9] in a practical application, both algorithms were tested on a collection of graphs that arise naturally in the computation of expansivity estimates in a dynamical system, studied in [5,7]. Graphs in a wide range of sizes were considered, with the number of vertices ranging from 1,000 to 16,000.…”

“…The author expresses his gratitude to Stefano Luzzatto for his encouragement to do extensive computations of expansion exponents in the quadratic map family for the paper [7], which motivated the development of the improved minimum cycle mean algorithm.…”

“…In the specific implementation used in [5], the total system memory of 256 GB was exhausted when processing graphs with somewhat over 125,000 vertices, while the computation time did not exceed 24 hours, thus encouraging to try processing even larger graphs in order to achieve better estimates of the expansion. The improved algorithm, introduced in Section 3 below, considerable decreased memory usage and allowed for the extensive computations described in [7].…”

Countable Markov partitions suitable for thermodynamic formalism

“…As an example let us consider several rectangles with indices starting from 1. It follows from the definition that [11] is admissible and [1i], i > 1 are not. Then R [11] is the only defining rectangle of order two for the As each index is at least 1 we get from the definition of admissible rectangles that inadmissible indices satisfy i N > N. Geometric condition G3…”

“…The main difficulty in that direction is to design a set of checkable numeric estimates which can be maintained through the induction. In the one-dimensional case such estimates were used in [22], [15], [11].…”