Geometrical Properties of the Mean-Median Map

TL;DR

This paper investigates the mean-median map as a dynamical system on piecewise-affine functions, analyzing the structure of its limit function near rational points and conjecturing about its Hausdorff dimension.

Contribution

It extends previous work by constructing a simplified map to analyze local dynamics, providing deeper insights into the structure of the limit function.

Findings

Determined the structure of the limit function near local minima.

Extended previous results by two orders of magnitude.

Conjectured the Hausdorff dimension of the graph exceeds 1.

Abstract

We study the mean-median map as a dynamical system on the space of finite sets 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 set $[0,x,1]$ is greater than 1.