Geometrical properties of the mean-median map

IF 1 Q3 Engineering

Journal of Computational Dynamics Pub Date : 2018-06-26 DOI:10.3934/jcd.2020004

Jonathan Hoseana, F. Vivaldi

引用次数: 3

Abstract

We study the mean-median map as a dynamical system on the space of finite multisets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive family of rational points, the local minima. By constructing a simpler map which represents the dynamics in such neighbourhoods, we extend the results of Cellarosi and Munday (arXiv:1408.3454v1 [math.CO]) by two orders of magnitude. Based on these computations, we conjecture that the Hausdorff dimension of the graph of the limit function of the multiset $[0,x,1]$ is greater than 1.

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中位数图的几何性质

我们将中值映射作为一个动力系统，在有限多集的有理系数分段仿射连续函数空间上进行研究。我们确定了一组特殊有理点的邻域极限函数的结构，即局部极小值。通过构造一个更简单的图来表示这样的邻域中的动态，我们将Cellarosi和Munday (arXiv:1408.3454v1 [math.CO])的结果扩展了两个数量级。基于这些计算，我们推测多集$[0,x,1]$的极限函数的图的Hausdorff维数大于1。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Computational Dynamics Engineering-Computational Mechanics

CiteScore

2.30

自引率

10.00%

发文量

期刊介绍： JCD is focused on the intersection of computation with deterministic and stochastic dynamics. The mission of the journal is to publish papers that explore new computational methods for analyzing dynamic problems or use novel dynamical methods to improve computation. The subject matter of JCD includes both fundamental mathematical contributions and applications to problems from science and engineering. A non-exhaustive list of topics includes * Computation of phase-space structures and bifurcations * Multi-time-scale methods * Structure-preserving integration * Nonlinear and stochastic model reduction * Set-valued numerical techniques * Network and distributed dynamics JCD includes both original research and survey papers that give a detailed and illuminating treatment of an important area of current interest. The editorial board of JCD consists of world-leading researchers from mathematics, engineering, and science, all of whom are experts in both computational methods and the theory of dynamical systems.