On the mass center of the tent map

TL;DR

This paper investigates the properties of exceptional orbits in the standard tent map, revealing the distribution and density of their mass centers, including periodic orbits and those without a mass center.

Contribution

It demonstrates the existence of multiple periodic orbits sharing the same mass center and characterizes the set of all possible mass centers for orbits in the tent map.

Findings

Set of mass centers of all orbits is [0, 2/3]

Dense subset of periodic orbit mass centers in [0, 2/3]

Uncountably many orbits without a mass center

Abstract

It is well known that the time average or the center of mass for generic orbits of the standard tent map is 0.5. In this paper we show some interesting properties of the exceptional orbits, including periodic orbits, orbits without mass center, and orbits with mass centers different from 0.5. We prove that for any positive integer $n$, there exist $n$ distinct periodic orbits for the standard tent map with the same center of mass, and the set of mass centers of periodic orbits is a dense subset of $[0,2/3]$. Considering all possible orbits, then the set of mass centers is the interval $[0,2/3]$. Moreover, for every $x$ in $[0,2/3]$, there are uncountably many orbits with mass center $x$. We also show that there are uncountably many orbits without mass center.