The work of George Szekeres
on
functional equations
Keith Briggs
Keith.Briggs@bt.com
http://keithbriggs.info
QMUL Maths 2006 January 10 16:00
corrected version 2006 January 12 13:41
Szekeres seminar QMUL 2006jan10.tex typeset 2006 January 12 13:41 in pdfLATEX on a linux system
The work of George Szekeres on functional equations
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George (György) Szekeres 1911-2005
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George Szekeres
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studied chemical engineering at Technological
University of Budapest
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refugee in Shanghai
1940s
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Adelaide 1948-63
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UNSW 1963-75
F
died Adelaide
2005 Aug 28
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The work of George Szekeres
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first co-author of Erd®s
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graph theory
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general relativity
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functional equations
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multi-dimensional continued fractions
F
lots more . . .
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Szekeres and colleagues
Paul Erdős, Esther Klein, George Szekeres, Fan Chung
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A letter from George
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Outline
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analytic iteration
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Schröder & Koenigs
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Szerekes on the Schröder and Abel equations
F
Szerekes on the Feigenbaum functional equation
F
Szerekes on Abel's equation and growth rates
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formal iteration & Julia's equation (my speculations)
F
Jacobian conjecture (my speculations)
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Iteration theory and functional equations
F
map f : X → X
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orbit xn = f (xn−1) ≡ f <n>(x0), n = 1, 2, 3, . . .
F
around 1870 Schröder proposed studying the orbit by trying to find
a new coordinate system in which the orbit `looks simpler'
F
simplest case: explicit iterability
F
σ◦f (x)−f 0(0)σ(x) = 0
F
σ◦f <n>(x) = (f 0(0))nσ(x),
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∀x
n = 0, 1, 2, . . .
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Friedrich Wilhelm Karl Ernst Schröder 1841-1902
F
Pforzheim
F
Ueber iterirte Functionen Math. Annalen 3 296-322 (1871)
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Solutions of the Schröder equation
F
Schröder found several explicit solutions to his equation in terms
of elementary functions
F
f1(x) =
F
F
F
2(x+x2) ⇒ σ(x) = log(1+2x)/2
√
2
f2(x) = −2(x+x ) ⇒ σ(x) = 3/2(arccos(−1/2−x)−2π/3)
√ 2
2
f3(x) = 4(x+x ) ⇒ σ(x) = (arcsinh( x))
cases 2 and 3 are conjugate with respect to the function
h(x) = −3/2−2x. That is, h◦f3 = f2 ◦h
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The Schröder equation
f
X −
−−−→ 
X



σy
yσ
Y −
−−−→ Y
x 7→f1 x
F
σ◦f (x) = f1σ(x) (Schröder, f1 ≡ f 0(0))
F
α◦f (x) = α(x)+1 (Abel)
F
β ◦f (x) = (β(x))2 (Boettcher=Bëther)
F
ι◦f (x) = f 0(x)ι(x) (Julia)
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F
Formal solution of the Schröder equation
P∞
P∞
i
for f (z) = i=1 fi z and σ(z) = i=1 σi z i we have

0
0
0
 f2 f12 −f1
0
 f 3f f f 3 −f
3
1 2
1
1
..
..
..
···
σ1
0
· · ·   σ2   0 
=
· · ·   σ3   0 
..
..
···




F
defn: family of (continuous) iterates: f <s>(z) = σ <−1>(f1sσ(z))
F
obtain formal solution from f <s> ◦f = f ◦f <s>
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Schröder, Abel, Boettcher, and Julia
f
f
X −
−−−→ 
X



σy
yσ
X −−−−→ X
X −
−−−→ 
X



αy
yα
X −
−−−−→ X
f
f
x 7→ f1 x
X −
−−−→ 
X



βy
yβ
X −
−−−→ X
x 7→ x2
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x 7→ x+1
X


ιy
X
−−−−→
−−−−−
−
→
0
x 7→ f (x) x
X

ι
y
X
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Gabriel Koenigs 1858-1931
F
F
F
F
F
F
let f (z) =
P∞
i=1
fi z i be analytic with |f1| < 1
then the
equation has an analytic solution
PSchröder
∞
σ(z) = i=1 σi z i, where each σk depends only on fi for i 6 k
also κ(z) ≡ limi→∞ f1−if <i>(z) exists and satisfies
κ◦f (z) = f1κ(z)
Kneser: sufficient to have f (z) = f1 z+O(|z|1+δ ), δ > 0 as z → 0
2
π
, σ has `flat spots'
Szekeres [1]: for f (z) = z2 + 3zπ sin |z|
Szekeres: if f : R → R is continuous, strictly monotone increasing,
f 0(x) exists and f 0(x) = a+O(xδ ), then the Koenigs function κ
exists and is invertible
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Solution of the Schröder equation [1]
F
cases for f :
. f (z) =
P∞
Pi=m
∞
i=m
fi z i analytic, m > 0
. f (z) ∼
fi z i as z → 0, m > 0
. f a continuous real function
F
case f1 6= 0, |f1| =
6 1: formal series exists and converges;
continuous iterates are analytic at 0
F
case |f1| = 1:
. f1 a root of unity: no formal solution
. f1 not a root of unity: convergence depends on arithmetic conditions; e.g. Siegel
log |f1n −1| = O(log(x)) as n → ∞ sufficient
. Baker f (z) = exp(z)−1: f <s> exists formally but diverges unless n ∈ Z
. Szekeres: there exists exactly one f <s> of which the formal series is an
asymptotic expansion as z → 0
. idea: use Abel equation as σ(z) = exp(α(z)), α(z) ∼ −1/z
. led to work by Écalle (Borel summability), Milnor (precise formulation of linearizability) and others
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The Feigenbaum functional equation 1
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consider scaling in the sequence of period-doubling bifurcations of
maps x 7→ µ−xn: we have orbit scaling α and parameter scaling δ
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solve for f : f (x) = γ −1f ◦f (γx), γ ≡ α−1 = f (1)
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f (b) = 0, f (x) ∼ c(b−x)n for x near b
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there exist such regular solutions for each n > 1
1.0
0.9
0.8
0.7
0.6
f2(x)0.5
0.4
0.3
0.2
0.1
0.0
0.0
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0.2
0.4
x
0.6
0.8
1.0
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The Feigenbaum functional equation 2
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what about cases where f → 0 faster than any power of x near b?
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Szekeres' idea [8]: convert to coupled system, one of which is a
Schröder or Abel equation:
. regular case:
2
f ◦h(x)
=
γ f (x)
h(x)
=
f (γf (x))
. singular case: set A(x) ∝ log(f (b−x)) - we get A(x) = c−2/x2 +c−1/x+
c0 log x+c+c1x+c2x2 +· · ·
F
the singular series are divergent but Borel summable: we get
γ∞ = 0.0333810598 . . . and δ∞ = 29.576303 . . .
F
Briggs & Dixon also solved the circle map case [8,9]
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g◦g(2x) = g(x),
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= g(1), obtaining ∞ = −0.275026971 . . .
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Abel's equation and regular growth
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we work here with real functions on [0, ∞)
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Abel: α◦f (x) = α(x)+1
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Abel: if α and α1 are strictly increasing C 1 solutions, then
α(x)−α1(x) = ψ◦α(x), where ψ is 1-periodic
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Lévy: which is the `best' solution?
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let c > 1, Cc be the set of strictly convex analytic functions with
f (0) = 0, f 0(0) = c, f 00(x) > 0, C = ∪cCc
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principal
function (best behaviour at 0): for
PAbel
∞
f (x) = i=1 fi xi ∈ Cc:
. α(x) = logc(x)+O(x) if f1 = c > 1
. α(x) = − f1x log(x)+O(x) if f1 = c = 1
2
F
we have thus selected a solution by its behaviour at 0. Szekeres
wants to study the behavior at ∞
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More functional equations
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D(x) ≡ 1/α0(x) satisfies D◦f (x) = f 0(x)D(x) (Julia)
P∞
<k>0
t(x) ≡ k=0 1/f
(x)
F
t satisfies t◦f (x) = f 0(x)(t(x)−1)
F
if f ∈ Cc, c > 1
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. α(x) = logc(x)+O(x)
”
“
f2
2
. D(x) = log(c) x+ c(c−1) x +. . .
. E ◦f (x) = f 0(x)(E(x)+D 0(x))
`
´
1
(t(x)−1)D 0(x)−t0(x)D(x)+E(x)
. φ(x) = D(x)
F
if f ∈ C1
.
.
.
.
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α(x) = −1/(f2x)+O(log(x))
D(x) = f2x2 +(f3 −f22)x3 +. . .
E ◦f (x) = f 0(x)E(x)+D 0 ◦f (x)
`
´
1
φ(x) = D(x)
t(x)D 0(x)−t0(x)D(x)+E(x)
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Yet more functional equations
∀x > 0
F
φ◦f (x) = φ(x)
F
ψ(x) ≡ φ(α<−1>(x)) is 1-periodic, if α is the principal Abel
function of f
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Szekeres' regularity criterion:
. ψ̂n ≡
R1
0
exp(2πins) ψ(s)ds = exp(αn +2πiβn)
. this defines a mapping (LF sequence) (C) → real-valued sequences αn
. defn: a sequence is completely monotonic if all differences of all orders are
non-negative
. defn: a sequence is L-regular if its first differences are the difference of two
completely monotonic bounded sequences
R1
. equivalent to being a moment sequence: αn = 0 tndχ(t)
. defn: f is regularly growing if its LF sequence is L-regular
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Szekeres' experimental results
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f (x) = 2x+x2: L-regular
e2
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f (x) = (1+x) −1: probably L-regular
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f (x) = 3x+x2: not L-regular
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f (x) = exp(cx)−1: probably L-regular
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f (x) = exp(x)+x−1: not L-regular
F
much more work needed to verify and extend these results!
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F
F
F
F
F
Some speculation by KMB on Julia's equation
P∞
0
ι◦f = f (x)ι(x) for formal series f (z) = i=1 fi z i
3
3 3
5
2 2
if f2 6= 0, then ι = (f3 −f2 )x + 2 f2 +f4 − 2 f3f2 x +· · ·
Julia inverse problem: any such series ι is formally conjugate to
ωa,b ≡ ax(xk +bx2k ) for appropriate a, b
we can thus solve the ODE (for k = 1): v (x) =
0
v 2(1−bv )
x2(1−bx)
The exact solution of this is explicitly

 1+W [+ exp{log(−b+ x1 )+( x1 −c)/b−1}/b] x <
−1
1
x=
bv(x) =

1+W [− exp{log(+b− x1 )+( x1 −c)/b−1}/b] x >
1
b
1
b
1
b
where W is Lambert's W function (W (z) exp(W (z)) = z )
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More speculation by KMB on Julia's equation
F
F
Jacobian conjecture: a polynomial map f : Rn → Rn with Jacobian determinant det J(f ) equal to unity has a polynomial inverse
we can write formally f (z) = exp(ωD)z , where D is the gradient
operator and ω satisfies J(f )ω = ω ◦f (multi-dimensional Julia
equation)
F
the mapping f 7→ ω is a bijection (Labelle)
F
then f (z) = exp(−ωD)z (z ∈ Rn)
F
det J(f ) can be expressed in terms of ω only
F
we thus have a reformulation of the Jacobian conjecture: show
that for all appropriate ω , both exp(ωD)z and exp(−ωD)z are
polynomial
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References
[1] GS Regular iteration of real and complex functions Acta Math. 100 203-258 (1958)
[2] GS Fractional iteration of exponentially growing functions J. Aust. Math. Soc. 2
301-320 (1962)
[3] K W Morris & GS Tables of the logarithm of iteration of ex −1 J. Aust. Math. Soc. 2
321-333 (1962)
[4] GS Fractional iteration of entire and rational functions J. Aust. Math. Soc. 4 129-142
(1964)
[5] GS Infinite iteration of integral summation methods J. Analyse Math. 24 1-51 (1971)
[6] GS Scales of infinity and Abel's functional equation Math. Chronicle 13 1-27 (1984)
[7] GS Abel's equation and regular growth: variations on a theme by Abel Experimental
Math. 7 85-100 (1998)
[8] K M Briggs, T Dixon & GS Analytic solutions of the Cvitanovi¢-Feigenbaum and
Feigenbaum-Kadanoff-Shenker equations Internat. J. Bifur. Chaos 8 347-357 (1998)
[9] K M Briggs, T Dixon & GS Essentially singular solutions of Feigenbaum-type functional
equations in Statistical physics on the eve of the 21st century 65-77 World Sci. (1999)
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