Question

Consider the subshift of finite type Σ′ ⊂ Σ determined by the rules 0 may follow 1 and both 0 and 1 may follow 0 , as discussed in Section 11.4. Prove that periodic points for σ are dense in Σ′

Answer 1

Periodic points in the subshift of finite type Σ′ are dense because for any sequence in Σ′, a periodic sequence can be constructed to be arbitrarily close by repeating patterns that adhere to the defined rules of the subshift.

Step-by-step explanation:

Proving that periodic points for σ are dense in Σ′ involves showing that for any point in Σ′ and any ε>0, there is a periodic point within ε of that point. A subshift of finite type is defined by certain restrictions on which symbols can follow others in sequences. The rules given, where '0 may follow 1' and both '0 and 1 may follow 0', indeed define a subshift of finite type. To show density of periodic points, consider any point in Σ′ and an arbitrary sequence within it. As per the rules, we can construct a periodic sequence by repeating a pattern like 10 or 00 (since both 1 and 0 can follow 0) to any length we desire. Such a construction will yield a periodic point arbitrarily close to our original sequence, hence demonstrating that periodic points are dense in Σ′.