Definition 1.1. A metric space (X, d) is an ultrametric space if and only if d

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1 MATEMATIQKI VESNIK 68, 4 (2016), December 2016 orgnaln nauqn rad research paper LOCALLY CONTRACTIVE MAPS ON PERFECT POLISH ULTRAMETRIC SPACES Francs George Abstract. In ths paper we present a result concernng locally contractve maps defned on subsets of perfect Polsh ultrametrc spaces (.e. separable complete ultrametrc spaces). Specfcally, we show that a perfect compact ultrametrc space cannot be contaned n ts locally contractve mage, a corollary relatng ths result to mnmal dynamcal systems, and pose a conjecture for the general Polsh ultrametrc case. 1. Prelmnares A topologcal space X s completely metrzable f X admts a compatble complete metrc d, and separable f X contans a countable dense subset. A Polsh space s a separable completely metrzable topologcal space. Lkewse, a metrc space that s both separable and complete we wll call a Polsh metrc space. Alexander Kechrs provdes a thorough treatment of Polsh spaces n [6, Secton 3, p. 13], and we follow the notaton contaned n ths reference for the majorty of the paper. Ths paper focuses on a partcular subclass of Polsh spaces, namely perfect Polsh ultrametrc spaces. A topologcal space s perfect f and only f t contans no solated ponts. Defnton 1.1. A metrc space (X, d) s an ultrametrc space f and only f d has the property that for all x, y X, d(x, y) max{d(x, z), d(y, z)}. In ths case we call d an ultrametrc. Brefly, the dameter of a set A, wll be denoted dam(a), where dam(a) = sup{d(x, y) : x, y A}. The followng are some fundamental facts about ultrametrc spaces whch we wll state wthout proof, see [9, p. 58], [6, p. 35]. Proposton 1.2. For an ultrametrc space (X, d) and ball B(x, r) X wth center x and radus r, the followng holds for all x, y, z X and nonempty Y X, 2010 Mathematcs Subject Classfcaton: 37B05, 54H20, 54E40 Keywords and phrases: contractve map; Polsh space; ultrametrc space, R-tree; dynamcal system. 233

2 234 F. George () d(x, z) d(y, z), mples d(x, y) = max{d(x, z), d(y, z)}, () All balls are clopen sets, () If two balls meet, then one s contaned n the other, (v) If y B(x, r), then B(x, r) = B(y, r), (v) If a Y, then dam(y ) = sup{d(a, b) : b Y }. Recall that a topologcal space X s zero-dmensonal f and only f X s Hausdorff and has a bass consstng of clopen sets. Therefore, any Polsh space that admts a compatble ultrametrc s zero-dmensonal, see [6, 7.1]. Some examples of zero-dmensonal spaces that admt a ultrametrc compatble wth ther usual topology are the p-adc numbers Q p, the Cantor Space 2 ω, the Bare Space ω ω. A Lpschtz functon possesses a stronger form of contnuty, whch s defned below. Defnton 1.3. For metrc space (X, d), a functon f : X X s Lpschtz f and only f there s some constant α 0 such that d(f(x), f(y)) αd(x, y) for all x, y X. If 0 < α < 1, then f s a contractve map. There exst other defntons for contractve functons n the mathematcal lterature. For the nterested reader, a thorough comparson of varous defntons for contractve maps was ntated by B.E. Rhoades, see [8]. But, for the purposes of ths paper we have chosen a commonly used defnton. In partcular, ths defnton of a contractve map s fundamental to the Banach Fxed Pont Theorem, a theorem of Analyss, also known as the Banach Contracton Prncple [3]. A neghborhood (nbhd) of x, denoted N x, s an open set contanng the pont x of a topologcal space X. Defnton 1.4. Let (X, d) be a metrc space. Then a map f : X X s a locally contractve map, f and only f for each x X, there exsts a N x X such that for some α x (0, 1), d(f(u), f(v)) α x d(u, v) for all u, v N x. Notce, the radal nature of the dameter of a ball n an ultrametrc space allows us to consder a weaker verson of a locally contractve map for the remander of ths paper, see Proposton 1.2 tem (v). Defnton 1.5. Let (X, d) be a metrc space. Then a map f : X X s a local radal contracton, f and only f for each x X, there exsts a N x X such that for some α x (0, 1), d(f(x), f(u)) α x d(x, u) for all u N x. For clarty and convenence, we wll ntroduce the defnton of an f-contractve nbhd. Defnton 1.6. Let (X, d) be a metrc space, f : X X a local radal contracton, and let N x X be a nbhd of x for some x X. Then N x s an

3 Locally contractve maps 235 f-contractve nbhd of x, f and only f for some α x (0, 1) we have for all u N x, d(f(x), f(u)) α x d(x, u). Our next lemma demonstrates the effect a local radal contracton f has on compact f-contractve nbhds. Lemma 1.7. Let (X, d) be a perfect compact Polsh ultrametrc space, let the map f : X X be a local radal contracton, and N x X be a compact f- contractve nbhd for some x X. Then, for some α (0, 1), dam(f[n x ]) α dam(n x ). Proof. Let X be a perfect compact Polsh ultrametrc space, and let the map f : X X be a local radal contracton. Also, let N x X be a compact f- contractve nbhd for some x X. By Defnton 1.6, we know there exsts an α (0, 1) such that for all u N x, d(f(x), f(u)) αd(x, u). Snce f[n x ] s compact, by the contnuty of f, t follows from Proposton 1.2 that dam(n x ) = d(x, x 1 ) for some x 1 N x such that x 1 x, by perfectness, and dam(f[n x ]) = d(f(x), f(x 2 )) for some x 2 N x. So, snce N x s a f-contractve nbhd, we have that d(f(x)), f(x 2 )) αd(x, x 2 ) αd(x, x 1 ). Therefore, dam(f[n x ]) α dam(n x ). We wll now state a useful fact wthout proof (see [9, p ]). Theorem 1.8. [W. H. Schkhof] Let (X, d) be an ultrametrc space. If U X s a nonempty open set, then U can be parttoned nto balls. We now package Theorem 1.8 nto a form more convenent for use n the proofs to follow, and note that compact metrc spaces are necessarly Polsh. Lemma 1.9. Let (X, d) be a perfect compact Polsh ultrametrc space, and let f : X X be a local radal contracton. Then, there exsts a fnte partton of X nto f-contractve balls. Proof. (Sketch) Let (X, d) be a perfect compact ultrametrc space, and let f : X X be a local radal contracton. Let N f be a collecton of f-contractve nbhds correspondng to each x X. Ths collecton forms an open cover of X, and by compactness can be assumed to be fnte. By Theorem 1.8 we can create a partton of each nbhd n N f nto balls, and agan by compactness each of these parttons can be assumed to be fnte. Thus, by ultrametrc property 1.2() t s easy to see that we can form B a fnte partton of X nto f-contractve balls. 2. Man results The proof of our man theorem reles upon the concept of a generalzed tree or R-tree ntroduced by Gao and Shao [4]. Ths generalzes the concept of a descrptve set-theoretc tree on ω (see [6, Secton 2, Chapter 1]), by allowng the tree s level or depth to be ndexed by any countable set R wth lmt pont 0 nstead of just by ω.

4 236 F. George Ths s done through the ntroducton of a dstance set for a metrc space (X, d), that s the set R = {d(x, y) : x y X}. Dstance sets of separable ultrametrc spaces are countable, see J.D. Clemens [2] and C. Shao [10]. The set R + denotes the set of non-negatve real numbers. Defnton 2.1. [4] For countable R R +, (X, d) s an R-ultrametrc space f {d(x, y) : x y X} R. For the remander of ths secton, unless otherwse specfed, fx R R + to be countable wth 0 R, where R denotes the set of all lmt ponts of R. Defnton 2.2. [4] Let ω <R denote the set of all functons. where a R, such that the set u: [a, + ) R ω supp(u) = {b R [a, + ) : u(b) 0} s fnte. We call (ω <R, ) the full R-tree. For some b dom(u), we denote u b the functon u ([b, + ) R) ω <R. For u, v ω <R, u s an ntal segment of v, denoted u v, f there exsts b dom(v) such that u = v b. For every u ω <R the level or depth of u s defned by l(u) = nf dom(u) = mn dom(u). For T ω <R, f v T and u v mples u T, then we call T an R-tree, and T s pruned f for all u T, there exsts a proper extenson v of u. A branch of an R-tree T s a functon f ω R such that for all a R, f a = f ([a, + ) R) T, and u f f u = f a. We wll let [T ] denote the set of all branches, also known as the end space or body of T, [4]. Defnton 2.3. [4] Let T be an R-tree. Defne a metrc on [T ] by { 0, f = g d(f, g) = max{a R : f(a) g(a)}, otherwse Theorem 2.4. [4] ([T ], d) s a Polsh R-ultrametrc space. Theorem 2.5. [4] For any Polsh R-ultrametrc space (X, ρ) there s a R-tree T such that (X, ρ) s sometrc to ([T ], d). Defnton 2.6. [4] For any u ω <R defne N u = {f [ω <R ] : u f} We see above that any branch of an R-tree T restrcted to any element of of R s an element of T. Together wth the fact that for any perfect compact R- ultrametrc space, the set contanng all u T where l(u) = r for some r R s fnte (see [4, p. 500]), we show the exstence of at least one unform dameter fnte partton of [T ] nto balls, where unform dameter s meant n the sense that each

5 Locally contractve maps 237 ball s of equal dameter. Brefly, for any R-tree T and u, v T f u v or v u we wll call u and v compatble, and f ths s not the case, ncompatble, whch we wll denote by u v. Also, t s easy to show that f u v, then N u N v =. The followng result by J.D. Clemens s helpful n our next lemma and theorem. Theorem 2.7. [2] A set A [0, ) s a set of dstances of some perfect, compact, ultrametrc space f and only f A can be enumerated as a countable decreasng sequence d : 0 wth lm d = 0. Now, we defne a property of a dameter for use n Lemma 2.9. Defnton 2.8. Let B be a cover of a metrc space X. For r R, where R s the dstance set to X, r s an nfmum dameter of R f there exsts B 0 B such that for all B B, dam(b 0 ) = r dam(b). Lemma 2.9. Let ([T ], d) be a perfect compact R-ultrametrc space, where T s an R-tree. If R R + s countable and wth R 0, then [T ] has a unform dameter partton for each r R such that r s an nfmum dameter of R. Proof. Snce R s countable and R 0 there exsts an R-tree T such that ([T ], d) s an nfnte compact ultrametrc space. Therefore, we can refne any cover of [T ] to a fnte partton {N u : < n} where each u T for some n < ω. Wthout loss of generalty, we can assume ths partton s ordered n a non-decreasng fashon wth respect to level, and thus label l 0 = l(u 0 ). Notce, snce the set {u T : l(u) = l 0 } s fnte t follows that the set V = {u, v T : l(u) = l(v) = l 0 & u v} s fnte. The collecton C = {N u : u V } s clearly fnte, and C s a cover of [T ], snce for all f [T ], there exsts f u T where u V, mplyng f N u C, by the defnton of a branch of T. Then t follows that C s a fnte partton of [T ], snce for all u, v V u v mples N u N v = (see remarks followng Defnton 2.8. Now, snce R can be enumerated nto a strctly decreasng sequence r : 0, by Theorem 2.7, then t follows that for all u V, l(u) = l 0 = r j for some j ω > r j+1 = dam(n u ) followng from the defnton of a branch of T. Therefore t follows, that for all N, M C, dam(n) = dam(m), and l 0 nfmum dameter by Defnton 2.8. Now, we are ready to present the proof of our man theorem. s a Theorem Let (X, ρ) be a nonempty perfect Polsh ultrametrc space. If X s compact, then any local radal contracton ϕ: X X s necessarly not surjectve.

6 238 F. George Proof. Let (X, ρ) be a perfect compact Polsh ultrametrc space. Let R R + be such that R = {ρ(x, y) : x y X}. Snce ths s the dstance set of a compact perfect ultrametrc space, we see by 2.7 that R can be enumerated as a strctly decreasng sequence r j : j 0 and lm j r j = 0. Snce R s countable and R 0, then by Theorem 2.5 snce X s a Polsh R- ultrametrc space there exsts a unque pruned R-tree T such that (X, ρ) s sometrc to ([T ], d). So, let σ : X [T ] be an sometry, and defne τ = σ ϕ σ 1 : [T ] [T ]. It s easy to see that τ s local radal contracton. Thus, t suffces to show that τ s not surjectve. By Lemma 1.9 for some n ω, we have a fnte partton {N u : u T and < n} of τ-contractve balls n [T ]. We may assume that l(u ) l(u j ) whenever j. Label l 0 = l(u 0 ) (.e. the lowest level nbhd). Now for each < n, let {v k : k < s } be the set of all ncompatble extensons of u wth l(v k ) = l 0, whch we know s possble by Lemma 2.9. So, dam N v k = r t for some t < ω and dam τ[n v k ] r t+1 < r t for all < n and k < s. Notce for all < n, τ[n v k ] s contaned n some ball B k where dam Bk r t+1, and hence by Proposton 1.2() for some < n and k < s. τ[n v k ] B k N v k Snce {N v k : < n & k < s } s a fnte partton of [T ], t s easy to see that τ[[t ]] [T ]. Below are some corollares related to the above result. The generc element of a set has a partcular property f and only f the set of all elements wth ths property forms a comeager subset (.e. a subset whch s the complement of a countable unon of nowhere dense subsets). The hyperspace of compact sets of a topologcal space X, denoted K(X), s equpped wth the Vetors Topology generated by the sets, {K X : K U}, {K X : K U }, where U s open n X. It s well known that for a perfect Polsh space X the hyperspace K(X) s a Bare space (see [6, p. 41]), whch allows us to take advantage of mplcatons of the Bare Category Theorem. So, from [6, p. 42] we see that the set K p (X) = {K K(X) : K s perfect } s dense G δ, snce X s perfect Polsh. Thus, snce K(X) s a Bare space, K p (X) s comeager, and therefore we see that the generc compact subset of a perfect Polsh space s perfect. So, recallng the above theorem, snce all self-maps whch local radal contractons defned on a perfect compact ultrametrc space are not surjectve, then the generc compact set of a perfect Polsh ultrametrc space has ths property, whch s stated as a corollary below. Corollary Let (X, d) be a perfect Polsh ultrametrc space. Then for generc compact K X and local radal contracton ϕ: K K, K ϕ[k].

7 Locally contractve maps 239 Now leadng toward a orgnal motvaton of the study of ths phenomena concernng locally contractve maps, we state the followng corollary related to topologcal dynamcal systems, whch s a par X, ϕ, where X s a closed topologcal space, and ϕ: X X s a contnuous map. A set E X s ϕ-nvarant, f ϕ(e) E. A topologcal dynamcal system X, ϕ s mnmal f and only f X contans no nonempty closed proper ϕ-nvarant sets, and n ths case ϕ s called a mnmal map. Observe, that f ϕ s a mnmal map and X s compact Hausdorff, then ϕ s surjectve (see S. Kolyada and L. Snoha [7]), and from ths, the subsequent corollary follows. Corollary If (X, d) s a perfect compact ultrametrc space and X, ϕ s a topologcal dynamcal system wth ϕ: X X local radal contracton, then X, ϕ s not mnmal. In Corollary 2.12, the assumpton that X s ultrametrc s essental. K. Ceselsk and J. Jasnsk recently proved that there exsts an uncountable perfect compact subset X of R and a surjectve local radal contracton f : X X such that X, f s a mnmal dynamcal system, see [1]. Brefly, consderng the general Polsh ultrametrc case as a whole, t should be noted that any fnte space (ultrametrc or not) s necessarly dscrete and thus there exsts a trval locally contractve map (.e. the dentty map). Thus, t suffces for us to consder nfnte spaces only. Also, related to the ablty to employ an R-tree n ths characterzaton, any Polsh R-ultrametrc space such that R does not contan 0 s necessarly a countable dscrete space (see [10, p. 17 and Remark 3.5, p. 88]), and thus the dentty map agan can be used. So, together wth the fact that for any Polsh ultrametrc space R s countable, we can assume that there exsts an R-tree wth end space sometrc to the Polsh spaces we are consderng, and motvated by ths, the followng conjecture s posed. Conjecture Let (X, ρ) be an nfnte Polsh ultrametrc space. Then X s compact f and only f any locally contractve map ϕ: X X s necessarly not surjectve. Acknowledgement. I would lke to thank the referee for ther remarks, whch mproved the presentaton and clarty of ths paper. I would also lke to thank my faculty advsor Dr. Jakub Jasnsk for hs constant support and many helpful suggestons. Our dscussons on the paper and topcs surroundng my research project are what realzed ts potental and made me a better mathematcan. Part of ths research was supported by the Unversty of Scranton Presdent Summer Research Fellowshp of REFERENCES [1] K.C. Ceselsk and J. Jasnsk, An auto-homeomorphsm of a Cantor set wth zero dervatve everywhere, J. Math. Anal. Appl. 434 (2) (2016), [2] J.D. Clemens, The set of dstances n a Polsh metrc space, jclemens/preprnts/dst.pdf, preprnt (2007).

8 240 F. George [3] P.M. Ftzpatrck and H.L. Royden, Real Analyss, 4th ed., Prentce Hall, New York, [4] S. Gao and C. Shao, Polsh ultrametrc Urysohn spaces and ther sometry groups, Topology Appl. (2010), [5] G. Jungck, Local radal contractons a counterexample, Houst. J. Math. 8 (1982), [6] A. Kechrs, Classcal Descrptve Set Theory, Sprnger-Verlag, New York, [7] S. Kolyada and L. Snoha, Mnmal dynamcal systems, Scholarpeda, 4(11):5803, [8] B.E. Rhoades, A comparson of varous defntons of contractve mappngs, Trans. Amer. Math. Soc. 226, (1977), [9] W. H. Schkhof, Ultrametrc Calculus: An Introducton to p-adc Analyss, Cambrdge Unversty Press, Cambrdge, [10] C. Shao, Urysohn Ultrametrc Spaces and Isometry Groups, Ph. D. Thess, Unversty of North Texas Dgtal Lbrary, (receved ; n revsed form ; avalable onlne ) Department of Mathematcs, Unversty of Scranton, Scranton, PA USA E-mal: frank.j.george@gmal.com