We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982. A continuous map of an interval to the real line which sends the endpoints in opposite directions has a fixed point. We generalize this to maps on non-invariant continua in the plane under positively oriented maps of the plane (with appropriate boundary conditions). These methods imply that in some cases non-invariant continua in the plane are degenerate. This has important applications in complex dynamics. E.g., a special case of our results shows that if $X$ is a non-separating invariant subcontinuum of the Julia set of a polynomial $P$ containing no fixed Cremer points and exhibiting no local rotation at all fixed points, then $X$ must be a point. It follows that impressions of some external rays to polynomial Julia sets are degenerate.
Deep Dive into Fixed point theorems in plane continua with applications.
We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982. A continuous map of an interval to the real line which sends the endpoints in opposite directions has a fixed point. We generalize this to maps on non-invariant continua in the plane under positively oriented maps of the plane (with appropriate boundary conditions). These methods imply that in some cases non-invariant continua in the plane are degenerate. This has important applications in complex dynamics. E.g., a special case of o
3.2 Bell's Lollipop.
3.3 var(f, A) = -1 + 1 -1 = -1.
4.1 Maximal balls have disjoint hulls.
6.1 The strip S from Lemma 6.1.2 6.2 Uniqueness of the negative outchannel.
7.1 Replacing the links [a n(1)-1 , a n(1) ], . . . , [a m(1)-1 , a m(1) ] by a single link [a n(1)-1 , a m(1) ]. 7.2 Illustration to the proof of Theorem 7.4.7.
7.3 Illustration to the proof of Lemma 7.4.9. 7.4 A general puzzle-piece ix Preface By a continuum we mean a compact and connected metric space and by a nonseparating continuum X in the plane C we mean a continuum X ⊂ C such that C \ X is connected. Our work is motivated by the following long-standing problem [Ste35] in topology.
Plane Fixed Point Problem: “Does a continuous function taking a nonseparating plane continuum into itself always have a fixed point?”
To give the reader perspective we would like to make a few brief historical remarks (see [KW91,Bin69,Bin81] for much more information).
Borsuk [Bor35] showed in 1932 that the answer to the above question is yes if X is also locally connected. Cartwright and Littlewood [CL51] showed in 1951 that a map of a non-separating plane continuum X to itself has a fixed point if the map can be extended to an orientation-preserving homeomorphism of the plane. It was 27 years before Harold Bell [Bel78] extended this result to the class of all homeomorphisms of the plane. Then Bell announced in 1982 (see also Akis [Aki99]) that the Cartwright-Littlewood Theorem can be extended to the class of all holomorphic maps of the plane. For other partial results in this direction see, e.g., [Ham51,Hag71,Bel79,Min90,Hag96,Min99].
In this memoir the Plane Fixed Point Problem is addressed. We develop and further generalize tools, first introduced by Bell, to elucidate the action of a fixed point free map (should one exist). We are indebted to Bell for sharing his insights with us. Some of the results in this memoir were first obtained by him. Unfortunately, many of the proofs were not accessible. Since there are now multiple papers which rely heavily upon these tools (e.g., [OT07,BO09,BCLOS08]) we believe that they deserve to be developed in a coherent fashion. We also hope that by making these tools available to the mathematical community, other applications of these results will be found. In fact, we include in Part 2 of this text new applications which illustrate their usefulness.
Part 1 contains the basic theory, the main ideas of which are due to Bell. We introduce Bell’s notion of variation and prove his theorem that index equals variation increased by 1 (see Theorem 3.2.2). Bell’s Linchpin Theorem 4.2.5 for simply connected domains is extended to arbitrary domains in the sphere and proved using an elegant argument due to Kulkarni and Pinkall [KP94]. Our version of this theorem (Theorem 4.1.5) is essential for the results later in the paper.
Building upon these ideas, we will introduce in Part 1 the class of oriented maps of the plane and show that it decomposes into two classes, one of which preserves and the other of which reverses local orientation. The extension from holomorphic to positively oriented maps is important since it allows for simple local perturbations of the map (see Lemma 7.5.1) and significantly simplifies further usage of the developed tools. In Part 2 new applications of these results are considered. A Zorn’s Lemma argument shows, that if one assumes a negative solution to the Plane Fixed Point Problem, then there is a subcontinuum X which is minimal invariant. It follows from Theorem 6.1.4 that for such a minimal continuum, f (X) = X. We recover Bell’s result [Bel67] (see also Sieklucki [Sie68], and Iliadis [Ili70]) that the boundary of X is indecomposable with a dense channel (i.e., there exists a prime end E t such that the principal set of the external ray R t is all of ∂X).
As the first application we show in Chapter 6 that X has a unique outchannel (i.e., a channel in which points basically map farther and farther away from X) and this outchannel must have variation -1 (i.e., as the above mentioned points map farther and farther away from X, they are “flipped with respect to the center line of the channel”).
The next application of the tools developed in Part 1 directly relates to the Plane Fixed Point Problem. We introduce the class of oriented maps of the plane (i.e., all perfect maps of the plane onto itself which are the compositions of monotone and branched covering maps of the plane). The class of oriented maps consists of two subclasses: positively oriented and negatively oriented maps. In Theorem 7.1.3 we show that the Cartwright-Littlewood Theorem can be extended to positively oriented maps of the plane.
These results are used in [BO09]. There we consider a branched covering map f of the plane. It follows from the above that if f has an invariant and fixed point free continuum Z, then f must be negatively oriented. We show in [BO09] that if, moreover, f is an oriented map of degree 2, then Z must
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