A FIXED POINT THEOREM FOR THE PSEUDO-CIRCLE

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1 A FIXED POINT THEOREM FOR THE PSEUDO-CIRCLE J. P. BOROŃSKI Abstract. Let f : C C be a self-map of the pseudo-crcle C. Suppose that C s embedded nto an annulus A, so that t separates the two components of the boundary of A. Let F : A A be an extenson of f to A (.e. F C = f). If F s of degree d then f has at least d 1 fxed ponts. Ths result generalzes to all plane separatng crcle-lke contnua. 1. Introducton A contnuum s a compact and connected space contanng at least two ponts. A contnuum X has the fxed pont property (abbrevated f.p.p.) f for any map f : X X there s an x X such that f(x) = x. In [6] O. H. Hamlton proved that the pseudo-arc and all other arc-lke contnua posses the f.p.p.. A contnuum s arc-lke (crcle-lke) f t s the nverse lmt of spaces homeomorphc to [0, 1] (homeomorphc to S 1 ). A contnuum s ndecomposable f t s not the unon of two proper subcontnua. A contnuum s heredtarly ndecomposable f each subcontnuum s ndecomposable. The pseudo-arc, frst constructed by B. Knaster n 1922 [11], s characterzed as a heredtarly ndecomposable arc-lke contnuum [2]. The pseudo-crcle, frst descrbed by R. H. Bng n 1951 [2], s a planar heredtarly ndecomposable crcle-lke contnuum whch separates the plane nto two components. It s topologcally unque [4] and s not homogeneous [3], [14]. Every proper subcontnuum of the pseudo-crcle s a pseudo-arc. The pseudo-crcle does not have the f.p.p.. In [7] M. Handel constructed an area preservng C dffeomorphsm of the plane wth the pseudo-crcle as a mnmal set (.e. nvarant and closed set that contans no other set wth these two propertes). However, J. Kennedy and J. T. Rogers, Jr. [10] showed that f h s a homeomorphsm of the pseudo-crcle wth an nvarant composant L (.e. h(l) = L) then h has a fxed pont n L. It s the purpose of the present paper to prove a new fxed pont theorem for the pseudo-crcle, motvated by the fact that any self-map of the unt 2000 Mathematcs Subject Classfcaton. Prmary 54F15, 54H25; Secondary 54H20. Key words and phrases. Fxed pont, pseudo-crcle, pseudo-arc, crcle-lke contnuum, Nelsen class. 1

2 2 J. P. BOROŃSKI crcle S 1 of degree d has d 1 fxed ponts [9]. Namely, let C be the pseudocrcle embedded n an annulus A n such a way that the wndng number of each crcular chan n the sequence of crooked crcular chans defnng C s 1. Any self-map of C extends to a self-map of A, snce C s a closed subset of the Absolute Neghborhood Retract A. Note that for any postve nteger d, C admts a self-map f that extends to a self-map F of A of degree d, by the result of J. Heath [8] (see also [5]). We wll prove the followng theorem. Theorem 1.1. Let f : C C be a self-map of the pseudo-crcle C. Suppose that F : A A s an extenson of f to A (.e. F C = f). If F s of degree d then f has at least d 1 fxed ponts. The man dea n our proof of Theorem 1.1 wll be to use the unversal coverng space (Ã, τ) of the annulus A. The am of usng the unversal cover s twofold. Frst, t s to make use of the known propertes of lftng classes of self-maps of A n Ã. Ths s a standard approach n the Nelsen fxedpont theory of compact connected polyhedra. For the class of arbtrary plane separatng contnua ths dea orgnates from [12], where K. Kuperberg studed fxed ponts of orentaton reversng planar homeomorphsms n nvarant contnua. Second, t s to unfold gven crcular chan of dsks coverng C to an nfnte lnear chan of dsks that covers the fber τ 1 (C) n Ã, and then to use arguments patterned on Hamlton s proof of f.p.p. for arc-lke contnua [6]. Ths approach s motvated by the fact that any Hausdorff two-pont compactfcaton of the unversal cover of the pseudo-crcle s homeomorphc to the pseudo-arc [1] (see also [13]). 2. Prelmnares Recall that for a compact connected polyhedron M wth the unversal coverng τ : M M, a Nelsen class of a map ψ : M M s the set τ({ x M : ψ( x) = x}), where ψ : M M s a lft of ψ to M. Nonempty Nelsen classes defne a partton of the fxed-pont set of ψ. It s known that a self-map of S 1 of degree d has d 1 Nelsen classes. Usng smlar arguments as for S 1 one can derve that A exhbts the same property and for completeness sake we wll now recall how the lftng classes are defned (cf. [9], p.618). Let τ : Ã A be a unversal cover of A, where we can assume that A = {(r, θ) R 2 : 1 r 2, 0 θ < 2π} n polar coordnates, Ã = {(r, θ) R 2 : 1 r 2}, and τ(r, θ) = (r, θ(mod 2π)). Let F : A A be a map of degree d, F be any lft of F, and p = (r, θ) be a fxed pont of F n A. The followng propertes hold. (1) τ 1 (p) = {(r, θ + 2πn) : n Z}, and F (τ 1 (p)) τ 1 (p),

3 A FIXED POINT THEOREM FOR THE PSEUDO-CIRCLE 3 (2) There s an nteger ( m[ F, p] such that for every (r, θ + 2πn) τ 1 (p) F (r, θ + 2πn) = r, θ + 2π(dn + m[ F ), p]), (3) F has a fxed pont n τ 1 (p) ff m[ F, p] = (1 d)n for some n Z, (4) f F has a fxed pont n τ 1 (p) then F (x, y) = F (x, y + 2π) s a lft of F wth a fxed pont n τ 1 (p) ff m[ F, p] mod(d 1). Therefore f k s mod(d 1) then fxed ponts of Fk and fxed ponts of Fs project to the same fxed ponts of F n A. On the other hand f k s mod(d 1) then fxed ponts of F k and fxed ponts of F s project onto dfferent fxed ponts of F n A. Ths determnes d 1 lftng classes of F correspondng to d 1 Nelsen fxed-pont classes. It wll be convenent to use the followng metrc (r, θ) (r, θ ) = r r + θ θ for both A and Ã. To avod confuson we wll ndcate n whch of the two spaces the dstance s taken by wrtng A or Ã. Note that lm n x n x o à = 0 mples lm n τ(x n ) τ(x o ) A = 0, and τ s a local sometry wth respect to the two metrcs. Suppose that U s a cover of a connected set H;.e. H U. Recall that the nerve of U, denoted by N(U), s an abstract smplcal complex that satsfes the followng: vertces of N(U) are the elements of U the smplces of N(U) are the fnte subcollectons {U 1,..., U q } of U such that U 1 U 2... U q. For an ɛ > 0 we call a cover U of H an ɛ-cover f the dameter of each element of U s less then ɛ. Recall that a contnuum X s crcle-lke f and only f for each ɛ > 0 there s a fnte ɛ-cover U of X by open sets such that the underlyng space of N(U) s homeomorphc to S 1. We shall call a cover V of H an nfnte chan f N(V) s connected, has nfntely (countably) many vertces, and each vertex s of degree 2. Equvalently, one can enumerate the elements of V by ntegers so that V V j ff j 1, for any V, V j V. We wll make use of the followng fact stated n [1], p.1147, Step 4, where the pseudo-crcle was embedded n the Möbus band (see also Fgure 1). Proposton 2.1. Let U ɛ be a fnte ɛ-cover of C by closed dsks, wth the underlyng space of N(U ɛ ) homeomorphc to S 1. Let τ 1 (U ɛ ) = {Un ɛ : n Z} consst of dsjont homeomorphc copes of U ɛ n Ã. Then, for suffcently small ɛ, V ɛ = {Un ɛ : n Z, U ɛ U ɛ } s an ɛ-cover of τ 1 (C) that s an nfnte chan.

4 4 J. P. BOROŃSKI Ã τ A Fgure 1. Lftng a crcular chan to an nfnte chan. 3. Proof of Theorem 1.1 Proof. (of Theorem 1.1) Fx d 1 and let F be a lft of F. It s enough to show that F has a fxed pont n τ 1 (C), what wll mply that there are d 1 fxed ponts of F n C (at least one for each Nelsen class). Frst notce that τ 1 (C) s connected, as t s the common boundary of τ 1 (K 1 ) and τ 1 (K 2 ), where K 1 and K 2 are the two components of A \ C (see [15], Theorem 1. () ). For every m, let U 1 m be an 1 m-cover of C by closed dsks such that the underlyng space of N(U ɛ ) s homeomorphc to S 1. By Proposton 2.1, U 1 m lfts to an nfnte chan V 1 m that s an 1 m -cover of τ 1 (C). Let V 1 m V 1 m V 1 m j = {V 1 m : Z} and enumerate elements of V 1 m so that j 1. Set A m = {x τ 1 (C) : [x V 1 m {x τ 1 (C) : x, F (x) V 1 m B m = {x τ 1 (C) : [x V 1 m {x τ 1 (C) : x, F (x) V 1 m & F (x) V 1 m j ] [ < j]} for some Z}, & F (x) V 1 m j ] [ > j]} for some Z}.

5 A FIXED POINT THEOREM FOR THE PSEUDO-CIRCLE 5 Notce that A m and B m are closed (cf. [6]). Let F (r, θ) = (φ(r, θ), ψ(r, θ)). Snce for any pont (r, θ) à and any nteger n the followng equalty holds ψ(r, θ + 2nπ) (θ + 2nπ) = [ψ(r, θ) θ] + 2nπ(d 1) t follows that the sgn of ψ(r, θ +2nπ) (θ +2nπ) depends only on whether n or n, and consequently A m and B m. Therefore A m B m, snce A m B m = τ 1 (C). For every m choose x m A m B m. We have F (x m ) x m à 1 m and thus lm m F (x m ) x m à = 0. Consequently also lm m F (τ(x m )) τ(x m ) A = 0. Notce that snce C s compact therefore there s c C such that lm m τ(x α(m) ) = c, for a subsequence {x α(m) } m=1 {x m} m=1. Clearly F (c) = c and therefore F (τ 1 (c)) τ 1 (c). We shall show that there s x o τ 1 (c) that s a fxed pont of F. There s a dsk D around c such that the dameter of G = D F (D) s less than 1 4. There s k > 4 such that τ(x α(k)), F (τ(x α(k) )) G. To fnsh the proof notce that τ 1 (G) = n Z G n conssts of dsjont homeomorphc copes of G. Let G k be the component of τ 1 (G) that contans x α(k). Note that snce F (x α(k) ) x α(k) à < 1 α(k) and the dameter of each G n s less than 1 4 we must have F (x α(k) ) G k. Now choose x o τ 1 (c) G k. It follows that F (x o ) τ 1 (c) G k and consequently F (x o ) = x o. Corollary 3.1. Let f : C C be a self-map of the pseudo-crcle C. Suppose that F : A A s an extenson of f to A (.e. F C = f). If F s of degree d then the fxed-pont set of f has at least d 1 components. Proof. It s known that each Nelsen class s an open subset of the fxedpont set (see [9], p.623). Snce there are only fntely many classes, each of them s also closed n the fxed pont set. The fxed-pont set of F s parttoned nto d 1 nonempty Nelsen classes N 1,..., N d 1. Let K be a component of the fxed-pont set of F n C and set M = N K. Snce each M s closed and K s connected t follows that only one M s nonempty. Remark 3.2. It should be clear from the proof of Theorem 1.1 that the result extends to all plane separatng crcle-lke contnua. Indeed, let K be such a contnuum separatng the two components of A, and let U ɛ be an ɛ-cover of K by open sets wth N(U ɛ ) homeomorphc to S 1. If τ 1 (U ɛ ) = {U ɛ n : n Z} and V ɛ = {U ɛ n : n Z, U ɛ U ɛ } then N(V ɛ ) s connected, as τ 1 (K) s. On the other hand, one easly verfes that, for suffcently small ɛ, every vertex of N(V ɛ ) s of degree 2, snce τ s a local homeomorphsm. As N(V ɛ ) has countably many vertces t follows that Proposton 2.1 holds for K. Now one can argue the same way as n the proof of Theorem 1.1.

6 6 J. P. BOROŃSKI Acknowledgments. The author would lke to thank professor Krystyna Kuperberg for many helpful comments that mproved the paper. References 1. Bellamy, D. P.; Lews, W., An orentaton reversng homeomorphsm of the plane wth nvarant pseudo-arc. Proc. Amer. Math. Soc. 114 (1992), no. 4, Bng, R.H. Concernng heredtarly ndecomposable contnua, Pacfc J. Math. 1 (1951), Fearnley, L. The pseudo-crcle s not homogeneous, Bull. Amer. Math. Soc. 75 (1969), Fearnley, L. The pseudo-crcle s unque, Trans. Amer. Math. Soc. 149 (1970), Gammon, K., Lftng crooked crcular chans to coverng spaces. Top. Proc., Vol. 36 (2010), Hamlton, O. H., A fxed pont theorem for pseudo-arcs and certan other metrc contnua. Proc. Amer. Math. Soc. 2, (1951), Handel, M., A pathologcal area preservng C dffeomorphsm of the plane. Proc. Amer. Math. Soc., 86 (1982), no. 1, Heath, J. W., Weakly confluent, 2-to-1 maps on heredtarly ndecomposable contnua. Proc. Amer. Math. Soc. 117 (1993), no. 2, Jang, B., A prmer of Nelsen fxed pont theory. Handbook of topologcal fxed pont theory, , Sprnger, Dordrecht, Kennedy, J.; Rogers, J. T., Jr. Orbts of the pseudocrcle. Trans. Amer. Math. Soc. 296 (1986), no. 1, Knaster, B. Un contnu dont tout sous-contnu est ndecomposable, Fund. Math. 3 (1922), Kuperberg, K., Fxed Ponts of Orentaton Reversng Homeomorphsms of the Plane. Proc. Amer. Math. Soc., Vol. 112, No. 1, (May, 1991), Kuperberg, K.; Gammon, K., A short proof of nonhomogenety of the pseudo-crcle. Proc. Amer. Math. Soc. 137 (2009), no. 3, Rogers, J.T. Jr., The pseudo-crcle s not homogeneous, Trans. Amer. Math. Soc. 148 (1970), Stone, A. H., Incdence relatons n uncoherent spaces. Trans. Amer. Math. Soc. 65, (1949) Department of Mathematcs, Auburn Unversty at Montgomery, P.O. Box Montgomery, AL Faculty of Appled Mathematcs, AGH Unversty of Scence and Technology, al. Mckewcza 30, Kraków, Poland E-mal address: boronjp@auburn.edu