Use Problem 11 to construct a continuous map that has fixed points of period 2n for all n ∈ Z+, but has no points of any other period. Problem 11 Let f be a map defined on the interval . Define ˜f,...

Use Problem 11 to construct a continuous map that has fixed points of period 2n for all n ∈ Z+, but has no points of any other period.

Problem 11

Let f be a map defined on the interval
. Define ˜f, “the double of f,” on
, as follows:

and filling the rest of the graph as in Fig. 2.28. Prove that ˜f has a 2nperiodic point at x if and only if f has an n-periodic point at x. Show that if f has points of period 2k(2n + 1), then ˜f has points of period 2k+1(2n + 1).

Fig. 2.28