From an old qualifying exam:
Assume that a particular electric switch has three states: left, middle, and right. Assume that the switch is programmed to follow these instructions at the start of each hour:
- If the switch was pointing to the left for the last hour then it flips to the right
- If the switch was pointing to the right for the last hour then it flips either to the left or the middle with 50% probability for each choice.
- If the switch was pointing to the middle for the last hour then it either stays where is is, or flips to the right (with probability 50% for each choice)
Now assume you initialize a very large number of independently operating switches of this type. Let the vector $s^{[0]}\in \mathbb{R}^3$ denote the distribution of switch states, such that $s_1^{[0]}, s_2^{[0]},s_3^{[0]}$ represent the fraction of the switches in the left, middle, and right states, respectively (note that $s_1^{[0]} + s_2^{[0]} + s_3^{[0]}$ must equal one.
Problem: characterize the state distributions that could not have arisen naturally from a previous state distribution.
My attempt so far: There is an answer attempt here (where it says "part 4 answer") that I do not find satisfactory. All admissible input vectors lie in the region $D$ between the coordinate planes and the triangle $T=\{x+y+z=1\mid x,y,z\geq0\}$. The answer's strategy is to note that the transition matrix $$L = \left( \begin{matrix} 0&0& 1/2 \\ 0& 1/2 & 1/2 \\ 1 & 1/2 &0 \end{matrix} \right)$$ has one eigenvalue $=1$ and the two other eigenvalues between zero and 1, so repeated applications of $T$ to a vector $v\in D$ will shrink it each time, eventually to $\rho_1v$, where $\rho_1$ projects onto the $1$ eigenspace along the other two eigenspaces. It says that the vectors in between $T$ and the image of $T$ under $L$ could not arise naturally from a previous state, and I agree with this. But there it stops. Surely there are more such vectors? In fact, all the vectors in $$D\setminus \{L^k(T) \mid k\in \mathbb{Z}^+\}$$ cannot arise naturally, right? Can we characterize them more precisely than that?
If $v$ is in $D$, not in $L^2(T)$, but is in $L(T)$, then it is $L(w)$ for some $w$ in $T$, so it does arise naturally from the previous state $w$.
You may be thinking, but you can't trace it any farther back than one iteration, but I think that's all the question is asking for.