Koenigs function

In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representation as dilations of a univalent holomorphic mapping, or a semigroup of mappings, of the unit disk in the complex numbers into itself.

Existence and uniqueness of Koenigs function

Let D be the unit disk in the complex numbers. Let Template:Mvar be a holomorphic function mapping D into itself, fixing the point 0, with Template:Mvar not identically 0 and Template:Mvar not an automorphism of D, i.e. a Möbius transformation defined by a matrix in SU(1,1).

By the Denjoy-Wolff theorem, Template:Mvar leaves invariant each disk |z | < r and the iterates of Template:Mvar converge uniformly on compacta to 0: in fact for 0 < Template:Mvar < 1,

$|f(z)|\le M(r)|z|$

for |z | ≤ r with M(r ) < 1. Moreover Template:Mvar '(0) = Template:Mvar with 0 < |Template:Mvar| < 1.

Template:Harvtxt proved that there is a unique holomorphic function h defined on D, called the Koenigs function, such that Template:Mvar(0) = 0, Template:Mvar '(0) = 1 and Schröder's equation is satisfied,

$h(f(z))=f^{\prime }(0)h(z).$

The function h is the uniform limit on compacta of the normalized iterates, $g_n(z)={\lambda }^{-n}{f}^n(z)$.

Moreover, if Template:Mvar is univalent, so is Template:Mvar.

As a consequence, when Template:Mvar (and hence Template:Mvar) are univalent, Template:Mvar can be identified with the open domain Template:Math. Under this conformal identification, the mapping   Template:Mvar becomes multiplication by Template:Mvar, a dilation on Template:Mvar.

Proof

• Uniqueness. If Template:Mvar is another solution then, by analyticity, it suffices to show that k = h near 0. Let

$H=k\circ h^{-1}(z)$

near 0. Thus H(0) =0, H'(0)=1 and, for |z | small,

$\lambda H(z)=\lambda h(k^{-1}(z))=h(f(k^{-1}(z))=h(k^{-1}(\lambda z)=H(\lambda z).$

Substituting into the power series for Template:Mvar, it follows that Template:Math near 0. Hence Template:Math near 0.

• Existence. If $F(z)=f(z)/\lambda z,$ then by the Schwarz lemma

$|F(z)-1|\le (1+|\lambda {|}^{-1})|z|.$

On the other hand,

$g_n(z)=z\prod _{j=0}^{n-1}F(f^j(z)).$

Hence g_n converges uniformly for |z| ≤ r by the Weierstrass M-test since

$\sum \underset{|z|\le r}{\sup }|1-F\circ f^j(z)|\le (1+|\lambda {|}^{-1})\sum M(r{)}^j<\mathrm{\infty }.$

• Univalence. By Hurwitz's theorem, since each gⁿ is univalent and normalized, i.e. fixes 0 and has derivative 1 there , their limit Template:Mvar is also univalent.

Koenigs function of a semigroup

Let Template:Math be a semigroup of holomorphic univalent mappings of Template:Mvar into itself fixing 0 defined for Template:Math such that

• $f_s$ is not an automorphism for Template:Mvar > 0
• $f_s(f_t(z))=f_{t+s}(z)$
• $f_0(z)=z$
• $f_t(z)$ is jointly continuous in Template:Mvar and Template:Mvar

Each Template:Math with Template:Mvar > 0 has the same Koenigs function, cf. iterated function. In fact, if h is the Koenigs function of Template:Math, then Template:Math satisfies Schroeder's equation and hence is proportion to h.

Taking derivatives gives

$h(f_s(z))=f_s^{\prime }(0)h(z).$

Hence Template:Mvar is the Koenigs function of Template:Math.

Structure of univalent semigroups

On the domain Template:Math, the maps Template:Math become multiplication by $\lambda (s)=f_s^{\prime }(0)$, a continuous semigroup. So $\lambda (s)=e^{\mu s}$ where Template:Mvar is a uniquely determined solution of Template:Math with ReTemplate:Mvar < 0. It follows that the semigroup is differentiable at 0. Let

$v(z)={\partial }_t{f}_t(z){|}_{t=0},$

a holomorphic function on Template:Mvar with v(0) = 0 and Template:Math = Template:Mvar.

Then

${\partial }_t(f_t(z))h^{\prime }(f_t(z))=\mu e^{\mu t}h(z)=\mu h(f_t(z)),$

so that

$v=v^{\prime }(0)\frac{h}{h^{\prime }}$

and

${\partial }_t{f}_t(z)=v(f_t(z)),f_t(z)=0,$

the flow equation for a vector field.

Restricting to the case with 0 < λ < 1, the h(D) must be starlike so that

$\mathrm{\Re }\frac{z{h}^{\prime }(z)}{h(z)}\ge 0.$

Since the same result holds for the reciprocal,

$\mathrm{\Re }\frac{v(z)}{z}\le 0,$

so that Template:Math satisfies the conditions of Template:Harvtxt

$v(z)=zp(z),\mathrm{\Re }p(z)\le 0,p^{\prime }(0)<0.$

Conversely, reversing the above steps, any holomorphic vector field Template:Math satisfying these conditions is associated to a semigroup Template:Math, with

$h(z)=z\exp \underset{0}{\overset{z}{\int }}\frac{v^{\prime }(0)}{v(w)}-\frac{1}{w}dw.$

References

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