Almost periodic homeomorphisms ofE²are periodic

Homeomorphisms of the plane

“…Further, as in [8], E f has no cut points. Thus it follows from Theorem 9 of [16], that the boundary of each of its complementary domains is a simple closed curve.…”

“…From this point on, the proof is exactly as in [8], if one replaces "almost periodic homeomorphism" by "homeomorphism with a family of equicontinuous iterates". 4* Dense orbits* Besicovitch in [4] and [5] gave an example of a homeomorphism of the plane such that the positive semi-orbit of some point is dense in E 2 .…”

“…By Theorem 2, there exists an invariant continuum K. We proceed as in the proof of Theorem 3.1 of [8]. Let ε > 0 and let {εj be a decreasing sequence of positive numbers such that Σεi < e. It follows from the equicontinuity of {T Now jDi is an invariant continuum, so for ε 2 there is d 2 > 0 such that diam T n (δ 2 -set) < ε 2 .…”

“…Continue the process inductively, and let E' = UΓ=i A Now E f is a locally connected continuum by Proposition 2.4 of [8], and is invariant. Further, as in [8], E f has no cut points.…”

“…This theorem was proved earlier by Kerekjartό [13], using different methods. Our proof of this uses ε-sequential growths and is similar to the proof of the main theorem of [8].…”