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. Rodrı́guez (Chachi)’s 60th birthday
Oviedo, 3-5 June 2015
Joint work with K.Barański, N.Fagella and B.Karpińska
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Newton maps
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Background
Definition: Let f be a polynomial or transcendental entire map. The map
N := Nf (z) = z −
f (z)
f 0 (z)
is called the Newton’s map associated to f .
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Newton maps
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Background
Definition: Let f be a polynomial or transcendental entire map. The map
N := Nf (z) = z −
f (z)
f 0 (z)
is called the Newton’s map associated to f .
If f is polynomial then Nf is rational.
If f is transcendental entire then (generally) Nf is transcendental
meromorphic.
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Newton maps
2 / 18
Background
Definition: Let f be a polynomial or transcendental entire map. The map
N := Nf (z) = z −
f (z)
f 0 (z)
is called the Newton’s map associated to f .
If f is polynomial then Nf is rational.
If f is transcendental entire then (generally) Nf is transcendental
meromorphic.
Newton’s method is one of the oldest and best known root-finding
algorithms.
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Newton maps
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Background
Definition: Let f be a polynomial or transcendental entire map. The map
N := Nf (z) = z −
f (z)
f 0 (z)
is called the Newton’s map associated to f .
If f is polynomial then Nf is rational.
If f is transcendental entire then (generally) Nf 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.
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Newton maps
2 / 18
Background
Definition: Let f be a polynomial or transcendental entire map. The map
N := Nf (z) = z −
f (z)
f 0 (z)
is called the Newton’s map associated to f .
If f is polynomial then Nf is rational.
If f is transcendental entire then (generally) Nf 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
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Newton maps
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Examples of Newton maps
f (z) = z(z − 1)(z − a), a ∈ C
→
Nf (z) = z −
z(z − 1)(z − a)
3z 2 − 2(1 + a)z + a
The phase portrait of the Newton map for a = 0.9091 + 0.415i.
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Newton maps
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Examples of Newton maps
f (z) = P(z) exp(z)
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→
Nf (z) = z −
Newton maps
P(z)
P(z) + P 0 (z)
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Examples of Newton maps
P(z) = z 10 − iz + 1
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→
NP (z) = z −
Newton maps
z 10 − iz + 1
10z 9 − i
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Examples of Newton maps
f (z) = exp (exp(−z))
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→
Newton maps
Nf (z) = z + exp(−z)
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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 ).
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Newton maps
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Illustration of Julia and Fatou sets
J (z 2 ) = ∂D
J (z + exp(−z))
(Cantor bouquet)
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Newton maps
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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.
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Newton maps
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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.
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Newton maps
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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.
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Newton maps
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Main Theorem and (direct) strategy proof
Main Theorem: Let f be a polynomial or an entire transcendental
function. Then, J (Nf ) is connected. Equivalently, all connected
components of the Fatou set are simply connected.
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Newton maps
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Main Theorem and (direct) strategy proof
Main Theorem: Let f be a polynomial or an entire transcendental
function. Then, J (Nf ) is connected. Equivalently, all connected
components of the Fatou set are simply connected.
Definition: Let f be a rational or transcendental function. Let z0 ∈ Ĉ a
fixed point of f , that is, f (z0 ) = z0 . We say that z0 is a weakly repelling
fixed point (WRFP) if |f 0 (z0 )| > 1 or f 0 (z0 ) = 1.
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Newton maps
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Main Theorem and (direct) strategy proof
Main Theorem: Let f be a polynomial or an entire transcendental
function. Then, J (Nf ) is connected. Equivalently, all connected
components of the Fatou set are simply connected.
Definition: Let f be a rational or transcendental function. Let z0 ∈ Ĉ a
fixed point of f , that is, f (z0 ) = z0 . We say that z0 is a weakly repelling
fixed point (WRFP) if |f 0 (z0 )| > 1 or f 0 (z0 ) = 1.
Remark: Let f be a polynomial or transcendental entire map, and let Nf
be its Newton’s map. Then Nf has no finite WRFP.
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Newton maps
10 / 18
Main Theorem and (direct) strategy proof
Main Theorem: Let f be a polynomial or an entire transcendental
function. Then, J (Nf ) is connected. Equivalently, all connected
components of the Fatou set are simply connected.
Definition: Let f be a rational or transcendental function. Let z0 ∈ Ĉ a
fixed point of f , that is, f (z0 ) = z0 . We say that z0 is a weakly repelling
fixed point (WRFP) if |f 0 (z0 )| > 1 or f 0 (z0 ) = 1.
Remark: Let f be a polynomial or transcendental entire map, and let Nf
be its Newton’s map. Then Nf has no finite WRFP.
Strategy of a DIRECT proof the Main Theorem ([BFJK,2015]): If there
was a multiply connected Fatou domain then Nf would have a finite
WRFP, a contradiction.
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Newton maps
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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 NP . Then, we define the basin of
attraction of α as
A(α) := {z0 ∈ C | lim NPn (z0 ) = α}
n→∞
Moreover, we denote by A? (α) ⊂ A(α) the immediate basin of attraction
of α, that it the component of A(α) containing α.
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Newton maps
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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 NP . Then, we define the basin of
attraction of α as
A(α) := {z0 ∈ C | lim NPn (z0 ) = α}
n→∞
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.
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Newton maps
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A nice application to polynomials
Definition: Let Pd be the space of polynomials of degree d, normalized so
that all their roots are in the open unit disk D.
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Newton maps
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A nice application to polynomials
Definition: Let Pd 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 ∈ Pd . Let {α1 , . . . αd } all roots
of P. Then there exists an explicit set Sd such that
#Sd ≈ 1.11d log2 d
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and Sd ∩ A? (αj ) 6= ∅, ∀j = 1, . . . , d.
Newton maps
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A nice application to polynomials
Definition: Let Pd 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 ∈ Pd . Let {α1 , . . . αd } all roots
of P. Then there exists an explicit set Sd such that
#Sd ≈ 1.11d log2 d
and Sd ∩ A? (αj ) 6= ∅, ∀j = 1, . . . , d.
The proof is based in the following considerations:
We can reduce to have all roots in D.
A? (αj ), j = 1 . . . d are unbounded.
A? (αj ), j = 1 . . . d are simply connected.
Size of the channels to infinity of the A? (αj )’s.
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Newton maps
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A nice application to polynomials
Example: Let P(z) = z 10 − iz + 1.
#Sd = 191 (≈ 1.11 × 10 × ln2 (10) ≈ 53).
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Newton maps
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This theorem has a long history: Rational case
f beging polynomial. Is J (Nf ) connected?
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Newton maps
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This theorem has a long history: Rational case
f beging polynomial. Is J (Nf ) connected?
Partial results from Przytycki ’86, Meier ’89, Tan Lei ...
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Newton maps
14 / 18
This theorem has a long history: Rational case
f beging polynomial. Is J (Nf ) connected?
Partial results from Przytycki ’86, Meier ’89, Tan Lei ...
A more general theorem on rational maps by Shishikura’90, closing
the problem.
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()
Newton maps
14 / 18
This theorem has a long history: Rational case
f beging polynomial. Is J (Nf ) 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.
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()
Newton maps
14 / 18
This theorem has a long history: Rational case
f beging polynomial. Is J (Nf ) 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 Nf be its associated (rational)
Newton’s map. Then J (Nf ) is connected.
K.B.; N.F. ; X.J. ; B.K.
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Newton maps
14 / 18
This theorem has a long history: Rational case
f beging polynomial. Is J (Nf ) 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 Nf be its associated (rational)
Newton’s map. Then J (Nf ) is connected.
Proof of the Corollary: Every rational map g has at least one WRFP
(Fatou’s Theorem). In case of g := Nf this WRFP is unique and located
at z = ∞.
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Newton maps
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This theorem has a long history: Transcendental case
f being transcendental entire (∞ essential singulary)
Is J (Nf ) connected?
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Newton maps
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This theorem has a long history: Transcendental case
f being transcendental entire (∞ essential singulary)
Is J (Nf ) connected?
A la Shishikura . . .
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Newton maps
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This theorem has a long history: Transcendental case
f being transcendental entire (∞ essential singulary)
Is J (Nf ) connected?
A la Shishikura . . .
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.
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Newton maps
15 / 18
This theorem has a long history: Transcendental case
f being transcendental entire (∞ essential singulary)
Is J (Nf ) connected?
A la Shishikura . . .
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.
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()
Newton maps
15 / 18
This theorem has a long history: Transcendental case
f being transcendental entire (∞ essential singulary)
Is J (Nf ) connected?
A la Shishikura . . .
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 (Nf ) is
connected.
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()
Newton maps
15 / 18
This theorem has a long history: Transcendental case
f being transcendental entire (∞ essential singulary)
Is J (Nf ) connected?
A la Shishikura . . .
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 (Nf ) is
connected.
Proof: If f is entire, Nf has no WRFP at all.
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Newton maps
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Final considerations
Shishikura’s proof (of the general theorem) and its extensions were
heavily based on surgery. The transcendental case was quite delicate.
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()
Newton maps
16 / 18
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
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
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
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 (Nf ) in all settings at once – rational and
transcendental; DIRECT – not as a corollary of the general result; and
therefore SIMPLER.
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()
Newton maps
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Some pictures I got... Fist time in Asturias
Ruta del Cares... (Who they are?)
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Newton maps
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Some pictures I got...
La Manga 1994
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UAB, 1997
Newton maps
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