Connectivity of the Julia set for Newton maps. Xavier Jarque (Universitat de Barcelona) Surfing the complexity A journey through Dynamical Systems

Transcription

1 Connectivity of the Julia set for Newton maps Xavier Jarque (Universitat de Barcelona) Surfing the complexity A journey through Dynamical Systems On the occasion of J. A. Rodríguez (Chachi) s 60th birthday Oviedo, 3-5 June 2015 Joint work with K.Barański, N.Fagella and B.Karpińska K.B.; N.F. ; X.J. ; B.K. () Newton maps 1 / 18

2 Background Definition: Let f be a polynomial or transcendental entire map. The map N := N f (z) = z f (z) f (z) is called the Newton s map associated to f. K.B.; N.F. ; X.J. ; B.K. () Newton maps 2 / 18

3 Background Definition: Let f be a polynomial or transcendental entire map. The map N := N f (z) = z f (z) f (z) is called the Newton s map associated to f. If f is polynomial then N f is rational. If f is transcendental entire then (generally) N f is transcendental meromorphic. K.B.; N.F. ; X.J. ; B.K. () Newton maps 2 / 18

4 Background Definition: Let f be a polynomial or transcendental entire map. The map N := N f (z) = z f (z) f (z) is called the Newton s map associated to f. If f is polynomial then N f is rational. If f is transcendental entire then (generally) N f is transcendental meromorphic. Newton s method is one of the oldest and best known root-finding algorithms. K.B.; N.F. ; X.J. ; B.K. () Newton maps 2 / 18

5 Background Definition: Let f be a polynomial or transcendental entire map. The map N := N f (z) = z f (z) f (z) is called the Newton s map associated to f. If f is polynomial then N f is rational. If f is transcendental entire then (generally) N f is transcendental meromorphic. Newton s method is one of the oldest and best known root-finding algorithms. It was one of the main motivations for the classical theory of holomorphic dynamics. K.B.; N.F. ; X.J. ; B.K. () Newton maps 2 / 18

6 Background Definition: Let f be a polynomial or transcendental entire map. The map N := N f (z) = z f (z) f (z) is called the Newton s map associated to f. If f is polynomial then N f is rational. If f is transcendental entire then (generally) N f is transcendental meromorphic. Newton s method is one of the oldest and best known root-finding algorithms. It was one of the main motivations for the classical theory of holomorphic dynamics. It defines a very interesting class of meromorphic maps: Those with NO FINITE NON-ATTRACTING FIXED POINTS K.B.; N.F. ; X.J. ; B.K. () Newton maps 2 / 18

7 Examples of Newton maps f (z) = z(z 1)(z a), a C N f (z) = z z(z 1)(z a) 3z 2 2(1 + a)z + a The phase portrait of the Newton map for a = i. K.B.; N.F. ; X.J. ; B.K. () Newton maps 3 / 18

8 Examples of Newton maps f (z) = P(z) exp(z) N f (z) = z P(z) P(z) + P (z) K.B.; N.F. ; X.J. ; B.K. () Newton maps 4 / 18

9 Examples of Newton maps P(z) = z 10 iz + 1 N P (z) = z z10 iz z 9 i K.B.; N.F. ; X.J. ; B.K. () Newton maps 5 / 18

10 Examples of Newton maps f (z) = exp (exp( z)) N f (z) = z + exp( z) K.B.; N.F. ; X.J. ; B.K. () Newton maps 6 / 18

11 Holomorphic dynamics: Phase space We divide the dynamical plane (phase space) in two completely invariant subsets: (a) The Fatou set: z Ĉ is in the Fatou set if f is normal at z. That is if there exist a neighborhood U of z such that f n U converge in compact subsets of U (to a holomorphic map or to infinity). We denote this set by F(f ). (b) The Julia set: The complement of F(f ) in Ĉ. We denote it by J (f ). Remark: If f is transcedental then J (f ). K.B.; N.F. ; X.J. ; B.K. () Newton maps 7 / 18

12 Illustration of Julia and Fatou sets J (z 2 ) = D J (z + exp( z)) (Cantor bouquet) K.B.; N.F. ; X.J. ; B.K. () Newton maps 8 / 18

13 Topological lemma Lemma/Definition: The Fatou set is open (and its complement, J (f ), is closed). Each connected component of F(f ) is called Fatou domain or Fatou component. They are either periodic (attracting basins, parabolic basins, Siegel discs, Herman rings or Baker domains), preperiodic or wandering. K.B.; N.F. ; X.J. ; B.K. () Newton maps 9 / 18

14 Topological lemma Lemma/Definition: The Fatou set is open (and its complement, J (f ), is closed). Each connected component of F(f ) is called Fatou domain or Fatou component. They are either periodic (attracting basins, parabolic basins, Siegel discs, Herman rings or Baker domains), preperiodic or wandering. Lemma: Let K is a compact subset of Ĉ. Then K is disconnected if and only if there is a Jordan curve γ such that K γ =, and K intersects the two connected components of Ĉ \ γ. K.B.; N.F. ; X.J. ; B.K. () Newton maps 9 / 18

15 Topological lemma Lemma/Definition: The Fatou set is open (and its complement, J (f ), is closed). Each connected component of F(f ) is called Fatou domain or Fatou component. They are either periodic (attracting basins, parabolic basins, Siegel discs, Herman rings or Baker domains), preperiodic or wandering. Lemma: Let K is a compact subset of Ĉ. Then K is disconnected if and only if there is a Jordan curve γ such that K γ =, and K intersects the two connected components of Ĉ \ γ. Corollary: J (f ) is connected in Ĉ if and only if each connected component of F(f ) is simply connected in C. K.B.; N.F. ; X.J. ; B.K. () Newton maps 9 / 18

16 Main Theorem and (direct) strategy proof Main Theorem: Let f be a polynomial or an entire transcendental function. Then, J (N f ) is connected. Equivalently, all connected components of the Fatou set are simply connected. K.B.; N.F. ; X.J. ; B.K. () Newton maps 10 / 18

17 Main Theorem and (direct) strategy proof Main Theorem: Let f be a polynomial or an entire transcendental function. Then, J (N f ) is connected. Equivalently, all connected components of the Fatou set are simply connected. Definition: Let f be a rational or transcendental function. Let z 0 Ĉ a fixed point of f, that is, f (z 0 ) = z 0. We say that z 0 is a weakly repelling fixed point (WRFP) if f (z 0 ) > 1 or f (z 0 ) = 1. K.B.; N.F. ; X.J. ; B.K. () Newton maps 10 / 18

18 Main Theorem and (direct) strategy proof Main Theorem: Let f be a polynomial or an entire transcendental function. Then, J (N f ) is connected. Equivalently, all connected components of the Fatou set are simply connected. Definition: Let f be a rational or transcendental function. Let z 0 Ĉ a fixed point of f, that is, f (z 0 ) = z 0. We say that z 0 is a weakly repelling fixed point (WRFP) if f (z 0 ) > 1 or f (z 0 ) = 1. Remark: Let f be a polynomial or transcendental entire map, and let N f be its Newton s map. Then N f has no finite WRFP. K.B.; N.F. ; X.J. ; B.K. () Newton maps 10 / 18

19 Main Theorem and (direct) strategy proof Main Theorem: Let f be a polynomial or an entire transcendental function. Then, J (N f ) is connected. Equivalently, all connected components of the Fatou set are simply connected. Definition: Let f be a rational or transcendental function. Let z 0 Ĉ a fixed point of f, that is, f (z 0 ) = z 0. We say that z 0 is a weakly repelling fixed point (WRFP) if f (z 0 ) > 1 or f (z 0 ) = 1. Remark: Let f be a polynomial or transcendental entire map, and let N f be its Newton s map. Then N f has no finite WRFP. Strategy of a DIRECT proof the Main Theorem ([BFJK,2015]): If there was a multiply connected Fatou domain then N f would have a finite WRFP, a contradiction. K.B.; N.F. ; X.J. ; B.K. () Newton maps 10 / 18

20 A nice application to polynomials Definition: Let P be a polynomial. Let α C be a zero of P and so a (super) attracting fixed point of N P. Then, we define the basin of attraction of α as A(α) := {z 0 C lim n Nn P (z 0) = α} Moreover, we denote by A (α) A(α) the immediate basin of attraction of α, that it the component of A(α) containing α. K.B.; N.F. ; X.J. ; B.K. () Newton maps 11 / 18

21 A nice application to polynomials Definition: Let P be a polynomial. Let α C be a zero of P and so a (super) attracting fixed point of N P. Then, we define the basin of attraction of α as A(α) := {z 0 C lim n Nn P (z 0) = α} Moreover, we denote by A (α) A(α) the immediate basin of attraction of α, that it the component of A(α) containing α. Remark: The study of the distribution and topology of the basins of attraction has produced efficient algorithms to locate all roots of P. K.B.; N.F. ; X.J. ; B.K. () Newton maps 11 / 18

22 A nice application to polynomials Definition: Let P d be the space of polynomials of degree d, normalized so that all their roots are in the open unit disk D. K.B.; N.F. ; X.J. ; B.K. () Newton maps 12 / 18

23 A nice application to polynomials Definition: Let P d be the space of polynomials of degree d, normalized so that all their roots are in the open unit disk D. Theorem (HSS, 2001) Fix d 2. Let P P d. Let {α 1,... α d } all roots of P. Then there exists an explicit set S d such that #S d 1.11d log 2 d and S d A (α j ), j = 1,..., d. K.B.; N.F. ; X.J. ; B.K. () Newton maps 12 / 18

24 A nice application to polynomials Definition: Let P d be the space of polynomials of degree d, normalized so that all their roots are in the open unit disk D. Theorem (HSS, 2001) Fix d 2. Let P P d. Let {α 1,... α d } all roots of P. Then there exists an explicit set S d such that #S d 1.11d log 2 d and S d A (α j ), j = 1,..., d. The proof is based in the following considerations: We can reduce to have all roots in D. A (α j ), j = 1... d are unbounded. A (α j ), j = 1... d are simply connected. Size of the channels to infinity of the A (α j ) s. K.B.; N.F. ; X.J. ; B.K. () Newton maps 12 / 18

25 A nice application to polynomials Example: Let P(z) = z 10 iz + 1. #S d = 191 ( ln 2 (10) 53). K.B.; N.F. ; X.J. ; B.K. () Newton maps 13 / 18

26 This theorem has a long history: Rational case f beging polynomial. Is J (N f ) connected? K.B.; N.F. ; X.J. ; B.K. () Newton maps 14 / 18

27 This theorem has a long history: Rational case f beging polynomial. Is J (N f ) connected? Partial results from Przytycki 86, Meier 89, Tan Lei... K.B.; N.F. ; X.J. ; B.K. () Newton maps 14 / 18

28 This theorem has a long history: Rational case f beging polynomial. Is J (N f ) connected? Partial results from Przytycki 86, Meier 89, Tan Lei... A more general theorem on rational maps by Shishikura 90, closing the problem. K.B.; N.F. ; X.J. ; B.K. () Newton maps 14 / 18

29 This theorem has a long history: Rational case f beging polynomial. Is J (N f ) connected? Partial results from Przytycki 86, Meier 89, Tan Lei... A more general theorem on rational maps by Shishikura 90, closing the problem. Theorem (Shishikura, 1991): Let g be any rational map. If J (g) is disconnected then g has, at least, 2 WRFP. K.B.; N.F. ; X.J. ; B.K. () Newton maps 14 / 18

30 This theorem has a long history: Rational case f beging polynomial. Is J (N f ) connected? Partial results from Przytycki 86, Meier 89, Tan Lei... A more general theorem on rational maps by Shishikura 90, closing the problem. Theorem (Shishikura, 1991): Let g be any rational map. If J (g) is disconnected then g has, at least, 2 WRFP. Corollary: Let f be a polynomial. Let N f be its associated (rational) Newton s map. Then J (N f ) is connected. K.B.; N.F. ; X.J. ; B.K. () Newton maps 14 / 18

31 This theorem has a long history: Rational case f beging polynomial. Is J (N f ) connected? Partial results from Przytycki 86, Meier 89, Tan Lei... A more general theorem on rational maps by Shishikura 90, closing the problem. Theorem (Shishikura, 1991): Let g be any rational map. If J (g) is disconnected then g has, at least, 2 WRFP. Corollary: Let f be a polynomial. Let N f be its associated (rational) Newton s map. Then J (N f ) is connected. Proof of the Corollary: Every rational map g has at least one WRFP (Fatou s Theorem). In case of g := N f this WRFP is unique and located at z =. K.B.; N.F. ; X.J. ; B.K. () Newton maps 14 / 18

32 This theorem has a long history: Transcendental case f being transcendental entire ( essential singulary) Is J (N f ) connected? K.B.; N.F. ; X.J. ; B.K. () Newton maps 15 / 18

33 This theorem has a long history: Transcendental case A la Shishikura... f being transcendental entire ( essential singulary) Is J (N f ) connected? K.B.; N.F. ; X.J. ; B.K. () Newton maps 15 / 18

34 This theorem has a long history: Transcendental case A la Shishikura... f being transcendental entire ( essential singulary) Is J (N f ) connected? Theorem (Bergweiler-Terglane 96,Mayer-Schleicher 06,Fagella-J-Taixés 08-11,Barański-Fagella-J-Karpińska 14 ): Let g be any transcendental meromorphic map. If J (g) is disconnected then g has, at least, 1 WRFP. K.B.; N.F. ; X.J. ; B.K. () Newton maps 15 / 18

35 This theorem has a long history: Transcendental case A la Shishikura... f being transcendental entire ( essential singulary) Is J (N f ) connected? Theorem (Bergweiler-Terglane 96,Mayer-Schleicher 06,Fagella-J-Taixés 08-11,Barański-Fagella-J-Karpińska 14 ): Let g be any transcendental meromorphic map. If J (g) is disconnected then g has, at least, 1 WRFP. Proof: It is done case by case (wandering, attracting basins, parabolic basins, Baker domains and Herman rings) using different techniques. K.B.; N.F. ; X.J. ; B.K. () Newton maps 15 / 18

36 This theorem has a long history: Transcendental case A la Shishikura... f being transcendental entire ( essential singulary) Is J (N f ) connected? Theorem (Bergweiler-Terglane 96,Mayer-Schleicher 06,Fagella-J-Taixés 08-11,Barański-Fagella-J-Karpińska 14 ): Let g be any transcendental meromorphic map. If J (g) is disconnected then g has, at least, 1 WRFP. Proof: It is done case by case (wandering, attracting basins, parabolic basins, Baker domains and Herman rings) using different techniques. Corollary: Let f be a transcendental entire function. Then J (N f ) is connected. K.B.; N.F. ; X.J. ; B.K. () Newton maps 15 / 18

37 This theorem has a long history: Transcendental case A la Shishikura... f being transcendental entire ( essential singulary) Is J (N f ) connected? Theorem (Bergweiler-Terglane 96,Mayer-Schleicher 06,Fagella-J-Taixés 08-11,Barański-Fagella-J-Karpińska 14 ): Let g be any transcendental meromorphic map. If J (g) is disconnected then g has, at least, 1 WRFP. Proof: It is done case by case (wandering, attracting basins, parabolic basins, Baker domains and Herman rings) using different techniques. Corollary: Let f be a transcendental entire function. Then J (N f ) is connected. Proof: If f is entire, N f has no WRFP at all. K.B.; N.F. ; X.J. ; B.K. () Newton maps 15 / 18

38 Final considerations Shishikura s proof (of the general theorem) and its extensions were heavily based on surgery. The transcendental case was quite delicate. K.B.; N.F. ; X.J. ; B.K. () Newton maps 16 / 18

39 Final considerations Shishikura s proof (of the general theorem) and its extensions were heavily based on surgery. The transcendental case was quite delicate. To conclude the problem, new tools were developed in [BFJK 14]: K.B.; N.F. ; X.J. ; B.K. () Newton maps 16 / 18

40 Final considerations Shishikura s proof (of the general theorem) and its extensions were heavily based on surgery. The transcendental case was quite delicate. To conclude the problem, new tools were developed in [BFJK 14]: Existence of absorbing regions inside Baker domains (as it is the case for attracting or parabolic basins). K.B.; N.F. ; X.J. ; B.K. () Newton maps 16 / 18

41 Final considerations Shishikura s proof (of the general theorem) and its extensions were heavily based on surgery. The transcendental case was quite delicate. To conclude the problem, new tools were developed in [BFJK 14]: Existence of absorbing regions inside Baker domains (as it is the case for attracting or parabolic basins). New strategy for the proof, different from all the previous ones, based on the existence of fixed points under certain situations. K.B.; N.F. ; X.J. ; B.K. () Newton maps 16 / 18

42 Final considerations Shishikura s proof (of the general theorem) and its extensions were heavily based on surgery. The transcendental case was quite delicate. To conclude the problem, new tools were developed in [BFJK 14]: Existence of absorbing regions inside Baker domains (as it is the case for attracting or parabolic basins). New strategy for the proof, different from all the previous ones, based on the existence of fixed points under certain situations. We now use this new tools to give a UNIFIED proof of the connectivity of J (N f ) in all settings at once rational and transcendental; DIRECT not as a corollary of the general result; and therefore SIMPLER. K.B.; N.F. ; X.J. ; B.K. () Newton maps 16 / 18

43 Some pictures I got... Fist time in Asturias Ruta del Cares... (Who they are?) K.B.; N.F. ; X.J. ; B.K. () Newton maps 17 / 18

44 Some pictures I got... La Manga 1994 UAB, 1997 K.B.; N.F. ; X.J. ; B.K. () Newton maps 18 / 18