On the mass center of the tent map

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.