A dynamical system (X,f) is called minimal if X does not contain any non-empty, proper, closed f-invariant subset. In such a case we also say that the map f itself is minimal. Thus, one cannot simplify the study of the dynamics of a minimal system by finding its nontrivial closed subsystems and studying first the dynamics restricted to them.

Exercise 2.2. Suppose F:RR is continuous and suppose there are closed ntervals I0,I1 such that F(I0)I1F(I1)I0. a. Show that F has at least one periodic orbit {x0,x1} of period q=2 such that x0I0 and x1I1. b. Show that if I0,I1 are disjoint, then such an orbit has minimal period q=2. c. Give an example to show that if I0I1=, then an orbit of minimal period 2 might not exist. Exercise 2.2. Suppose F:RR is continuous and suppose there are closed ntervals I0,I1 such that F(I0)I1F(I1)I0. a. Show that F has at least one periodic orbit {x0,x1} of period q=2 such that x0I0 and x1I1. b. Show that if I0,I1 are disjoint, then such an orbit has minimal period q=2. c. Give an example to show that if I0I1=, then an orbit of minimal period 2 might not exist