2

I'm having a hard time reading the Bruce Kitchen's book "Symbolic Dynamics-One-sided, Two-sided and Countable State Markov Shifts". In page 16, he introduces the Perron-Frobenius Theory with some basic knowledge, and I cannot understand them.

Here are the definitions:

A real, nonnegative, square matrix $A$ is called irreducible if for every pair of indices $i$ and $j$ there is an $l>0$ with $(A^l)_{ij}>0$. This is equivalent to saying that the related graph $G_A$ is strongly connected.

With the same condition on the matrix $A$, given an index $i$, let $p(i)=gcd\left\{ l:(A^l)_{ii}>0\right\}$. This is the period of the index i.

When A is irreducible the period of every index is the same and is called the period of A.

Then I have 2 questions on these.

  1. Is there any motivation for using gcd to be included in the definition of the period of the index? I found from Wikipedia that this definition is equivalent to the greatest common divisor of all lengths of the possible cycles at $i$. But why do we need gcd instead of the least length of the cycle?

  2. In the third paragraph of the definition box, I cannot found the reason for all periods of indices to be same for the irreducible matrix $A$. How can I prove this?

Community
  • 1
Jassenhaus
  • 153

1 Answers1

1

Consider a directed graph where vertex 1 is contained in a 2-cycle and a 3-cycle(e.g. 1→2→1 and 1→3→4→1) then every length of cycle can be expressed as $2m + 3n$ for some nonnegative integers $m,n$(not both zero). As you may see, if we allow $m,n$ to run through whole $\mathbb Z$ then $\{2m+3n\}$ is equal to $1\mathbb Z$. Even if we restrict them to be in $\mathbb N$(here $\mathbb N$ contains $0$), it remains that $1\mathbb N$ is the smallest cyclic submonoid of $(\mathbb N,+)$ including subsemigroup $\{2m + 3n\}$.

Generally sets $S_i = \{ l:(A^l)_{ii}>0\}$ form subsemigroups of $(\mathbb N,+)$, and $\gcd S_i$ is the generator of the smallest cyclic submonoid including $S_i$. Then, maybe it is not that weird to pick $\gcd S_i$ as a single number to represent $S_i$.

Free
  • 26
  • 2