On homeomorphisms and C 1 maps

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1 arxv: v1 [mah.gm] 7 Apr 018 On homeomorphsms and C 1 maps Nkolaos E. Sofronds Deparmen of Economcs, Unversy of Ioannna, Ioannna 45110, Greece. nsofron@oene.gr, nsofron@cc.uo.gr Absrac Our purpose n hs arcle s frs, followng [8], o prove ha f α, β are any pons of he open un dsc D0;1 n he complex plane C and r, s are any posve real numbers such ha Dα;r D0;1 and Dβ;s D0;1, hen here exs 0,1 and a homeomorphsm h [: D0;1 ] D0;1 such ha Dα;r D0;, Dβ;s D0;, h Dα; r = Dβ;s and h = d on D0;1 \ D0;, and second, followng [9], o prove ha f q N \ {0,1} and B0;1 s he open un ball n R q, whle for any > 0, we se f x = x 1+ 1 x, whenever x B0;1, hen f d n C 1 B0;1,R q as 1 +. Mahemacs Subjec Classfcaon: 03E65, 37E30, 54C35. 1 Inroducon Our purpose n hs arcle s o prove n ZF - Axom of Foundaon + Axom of Counable Choce one heorem regardng homeomorphsms of he closed un dsc D0;1 n he complex plane C and one heorem regardng maps n C 1 B0;1,R q, where B0;1 s he he closed un ball n R q and q N \ {0,1}. For he defnon of a homeomorphsm beween merc spaces Dedcaed o he memory ofmy grandparensnkolaosand Alexandra, and Konsannos and Elen. AΣMA : 130/543/94 1

2 see page 39 of [1] or page 84 of [] or page 18 of [3] or page 45 of [4] or page 144 of [6], whle he merc of C 1 B0;1,R q s easly deduced from pages 7-8 of [7] and pages of [5], snce f = f 1,...,f q C 1 B0;1,R q, f and only f for any {1,...,q}, we have ha f C 1 B0;1,R. A parcular homeomorphsm of D0; 1.1. Defnon. Leα D0;1bearbrarybufxedandleρ > 0besuch ha Dα;ρ D0;1 or equvalenly α + ρ < 1,.e., 0 < ρ < 1 α. Fnally, le 0 be such ha Dα;ρ+ D0;1 or equvalenly α +ρ+ < 1,.e., 0 < α ρ. Then, followng page 5 of [8], we se ψ α;ρ; α+re = α+ ρ+ ρ re f 0 r ρ, 0 < π α+ r ρ +ρ+ e f ρ r ρ+, 0 < π α+re oherwse whenever α + re D0;1, where r 0 and 0 < π. An almos verbam repeon of he argumen on page 5 of [8] proves ha he map [ 0, α ρ ψ α;ρ; HD0;1 s connuous, whle ψ α;ρ; = d on he un crcle... Defnon. Le a R and le 0 b < 1, whle 0 ǫ < 1 b. We se rexpθ+a f 0 r b, 0 θ < π σ a;b;ǫ re θ = rexp θ ar ǫ b+a f b r b+ǫ, 0 θ < π re θ f b+ǫ r 1, 0 θ < π I s no dffcul o verfy ha σ a;b;ǫ : D0;1 D0;1 consues a homeomorphsm whch s he deny on D0;1\D0;b+ǫ and roaon by a

3 on D0;b. Moreover, s no dffcul o verfy ha f 0 < < ǫ < 1 b, hen for any z D0;1, we have ha σ a;b;ǫ z σ a;b; z < ǫ..3. Defnon. Le 0 u < 1 be arbrary bu fxed and le > 0 be such ha [,u+] [,] D0;1. We consruc a homeomorphsm τ u; : D0;1 D0;1 whch s he deny on D0,1\,u+, and ranslaon by u on [u,u+] [,],.e., ranslaes [u,u+] [,] o [,]. We proceed by defnng τ u; on [,u+] [,]. If for any j {1,}, we se pr j : R x 1,x x j R, hen we dsngush he followng hree cases: y. If y x y, hen we se pr τ u; x,y = y, whle pr 1 τ u; x,y s defned as follows: a If u+ x u+, hen pr 1 τ u; x,y = 1 u+ y + u+ x u + u y +. b If u x u+, hen pr 1 τ u; x,y = x+ u y u. c If x u, hen pr 1 τ u; x,y = 1 u+ y + u+ x+. y. If y x y, hen we se pr τ u; x,y = y, whle u+x u + f u+ x u+ pr 1 τ u; x,y = x u f u x u+ u+ x+ f x u y. If y x y,henwesepr τ u; x,y = y, whle pr 1 τ u; x,y s defned as follows: 3

4 a If u+ x u+, hen pr 1 τ u; x,y = 1 u+ y ++ u+ x u + uy ++. b If u x u+, hen pr 1 τ u; x,y = x+ u y + u. c If x u, hen pr 1 τ u; x,y = 1 u+ y ++ u+ x+..4. Theorem. If α, β are any pons of D0;1 and r, s are any posve real numbers such ha Dα;r D0;1 and Dβ;s D0;1, hen here exs 0,1 and a homeomorphsm h : D0;1 D0;1 such ha Dα;r D0;, Dβ;s D0;, h [ Dα;r ] = Dβ;s and h = d on D0;1\D0;. Proof. If 0 < ǫ < 1 mn{ α,1 α, β,1 β }, hen s no dffcul 4 o verfy ha [ ] ψ α;r;ǫ r/ Dα;r f 0 < r < ǫ Dα;ǫ = [ ] ψ 1 α;ǫ;r ǫ/ Dα;r f 0 < ǫ < r and [ ] ψ β;s;ǫ s/ Dβ;s Dβ;ǫ = [ ] ψ 1 β;ǫ;s ǫ/ Dβ;s f 0 < s < ǫ f 0 < ǫ < s whle f 0 a < π and 0 b < π are such ha α = α e a and β = β e b, hen snce Dα;ǫ D0; α + ǫ and Dβ;ǫ D0; β + ǫ, s no dffcul o verfy ha [ f 0 < ] η < mn{1 α ǫ,1[ β ] ǫ}, hen D α ;ǫ = σ a; α +ǫ;η Dα;ǫ and D β ;ǫ = σ b; β +ǫ;η Dβ;ǫ. Therefore, he clam follows from he fac ha f ǫ s small enough for boh [ ǫ, α +ǫ] [ ǫ,ǫ], [ ] [ ǫ, β +ǫ] [ ǫ,ǫ] [ ] obeconanednd0;1, hen τ α ;ǫ D α ;ǫ = D0;ǫ = τ β ;ǫ D β ;ǫ. 4

5 3 Parcular maps n C 1 B0;1,R q 3.1. Defnon. If > 0, hen we se f x =.e., f f = f 1,...,f q, hen f x 1,...,x q = x 1+ 1 x, x 1+ 1 x x q whenever 1 q and x = x 1,...,x q B0;1. For q =, hs funcon s nroduced on page 1 of [9]. We remark ha f 1 = d, whle for any value of he parameer, a sraghforward compuaon shows ha f x j = x x q whenever he ndces, j are dsnc, and f x = x x j, 1+ 1 x xq 1+ 1 x 1 x x 1+ 1 x, for any ndex, where s no dffcul o prove ha he orgn consues a removable sngulary and of dsnc ndces, j. f x j x=0 3.. Lemma. f d 0 as 1. = 0 and f x x=0, =, for any par Proof. Gven any > 0 and any x B0;1, a sraghforward compuaon shows ha and hence max f x x = max f x x = 1 x x 1+ 1 x 1 x x 1+ 1 x 5 ss = 1 max 0 s s.

6 If > 1, hen obvously max 0 s 1 ss 1+ 1s max 0 s 1 ss = 1 and f d 1 0 as 1 +. So le us assume ha 0 < < 1. Then, a sraghforward compuaon shows ha d ss = s s+1 ds 1+ 1s 1+ 1s, where he roos of s s+1 = 0 are and 1+. Hence, snce follows mmedaely ha 0 < < 1 < 1+, d ds ss 1+ 1s and consequenly max 0 s 1 whch mples ha > 0 on [ 0, ss 1+ 1s = and d ss ds 1+ 1s < 0 on,1], 1+ 1 =, f d = = + 0 as 1 and he clam follows Lemma. For any ndex, f 1 x 0 as 1 +. Proof. If > 1, hen 6

7 f 1 x = max 1 x 1+ 1 x x 1+ 1 x + 1 max 1 x 1+ 1 x x 1+ 1 x max max 1+ 1 x max 1+ 1 x max 1+ 1 x max max 1+ 1 max x 1+ 1 = 1 x x 1+ 1 x x x 1+ 1 x x 1+ 1 x and he clam follows Lemma. For any par of dsnc ndces, j, we have ha f 0 as 1 +. x j Proof. If > 1, hen f = max x j x x j x x q 1+ 1 x x q x = max x j x 1+ 1 x = 1 max x x j x 1+ 1 x 7

8 = 1 max = 1 max = 1 max x 1 x x x 1+ 1 x x 1+ 1 x and he clam follows Theorem. f d n C 1 B0;1,R q as 1 +. Proof. I s an mmedae consequence of he prevous hree lemmas. References [1] C. D. Alprans and O. Burknshaw, Prncples of real analyss, Thrd Edon, Academc Press, [] T. M. Aposol, Mahemacal analyss, Second Edon, Addson Wesley, [3] Yu. Borsovch, N. Blznyakov, Ya. Izralevch, T. Fomenko, Inroducon o opology, Mr Publshers, Moscow, [4] J. Deudonné, Foundaons of modern analyss, Academc Press, New York and London, [5] K. Io, Edor, Encyclopedc dconary of mahemacs, Volume II, Second Edon, The MIT Press, Cambrdge, [6] H. L. Royden, Real analyss, Thrd Edon, Macmllan Publshng Company, New York, [7] N. E. Sofronds, Lecures on ndusral and appled mahemacs, Smmera Publcaons, 014. [8] N. E. Sofronds, Fxed pon free homeomorphsms of he complex plane, arxv, 9 Augus

9 [9] N. E. Sofronds, Dffeomorphsms of he closed un dsc convergng o he deny, arxv, 10 July