C defined by n(pl, x2) = 01/c1 (PI, X2). The function II is the rotation of the circle C by the irrational angle 27rC2/C1. There exists a positive integer k such that k/c1 > 1.
Define F:I18->Rby F=fop. 2. Recurrent Points 32 Consider the differential equation x = F(x) on R and its associated flow V satisfying gt(p(x)) = P((Dt(x)) for all real numbers x and t. By means of contradiction, assume that there exists 0 E S1 \Per(gt). Let x E P-'(0). Since f is positive, F is positive. Thus, bt(x) is an increasing function of t. We claim that (Dt (x) is bounded above as a function of t. In particular, we shall prove that V(x) - x < 1 for all t > 0. By means of contradiction, assume that there exists T > 0 such that (DT (x) - x > 1.
However, demonstrating that the periodic set satisfies the Restriction Property, and the Decomposition Property is straightforward. We include statements and proofs in the interest of completeness. 17. If Ot is a flow, then Per(Ot I per(ot)) = Per(u). Proof. If p E Per(Otlper(ot)), then p E Per(q5t). If p E Per(ot), then there exists a positive real number T such that O' (p) = p. Consequently, IPer(ot)(p) _ 0' (P) = pSo, p E Per(Otlper(ot)). Therefore, Per(otlper(ot)) = Per(u). 18. The periodic set of a flow is a union of disjoint closed sets which are invariant with respect to the flow.