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 f be a holomorphic function mapping D into itself, fixing the point 0, with f not identically 0 and f not an automorphism of D, i.e. a Möbius transformation defined by a matrix in SU(1,1).
By the Denjoy-Wolff theorem, f leaves invariant each disk |z | < r and the iterates of f converge uniformly on compacta to 0: in fact for 0 < r < 1,
for |z | ≤ r with M(r ) < 1. Moreover f '(0) = λ with 0 < |λ| < 1.
Koenigs (1884) proved that there is a unique holomorphic function h defined on D, called the Koenigs function, such that h(0) = 0, h '(0) = 1 and Schröder's equation is satisfied,
The function h is the uniform limit on compacta of the normalized iterates, .
Moreover, if f is univalent, so is h.[1][2]
As a consequence, when f (and hence h) are univalent, D can be identified with the open domain U = h(D). Under this conformal identification, the mapping f becomes multiplication by λ, a dilation on U.
Proof
- Uniqueness. If k is another solution then, by analyticity, it suffices to show that k = h near 0. Let
- near 0. Thus H(0) =0, H'(0)=1 and, for |z | small,
- Substituting into the power series for H, it follows that H(z) = z near 0. Hence h = k near 0.
- Existence. If then by the Schwarz lemma
- On the other hand,
- Hence gn converges uniformly for |z| ≤ r by the Weierstrass M-test since
- Univalence. By Hurwitz's theorem, since each gn is univalent and normalized, i.e. fixes 0 and has derivative 1 there , their limit h is also univalent.
Koenigs function of a semigroup
Let ft (z) be a semigroup of holomorphic univalent mappings of D into itself fixing 0 defined for t ∈ [0, ∞) such that
- is not an automorphism for s > 0
- is jointly continuous in t and z
Each fs with s > 0 has the same Koenigs function, cf. iterated function. In fact, if h is the Koenigs function of f = f1, then h(fs(z)) satisfies Schroeder's equation and hence is proportion to h.
Taking derivatives gives
Hence h is the Koenigs function of fs.
Structure of univalent semigroups
On the domain U = h(D), the maps fs become multiplication by , a continuous semigroup. So where μ is a uniquely determined solution of e μ = λ with Reμ < 0. It follows that the semigroup is differentiable at 0. Let
a holomorphic function on D with v(0) = 0 and v'(0) = μ.
Then
so that
and
the flow equation for a vector field.
Restricting to the case with 0 < λ < 1, the h(D) must be starlike so that
Since the same result holds for the reciprocal,
so that v(z) satisfies the conditions of Berkson & Porta (1978)
Conversely, reversing the above steps, any holomorphic vector field v(z) satisfying these conditions is associated to a semigroup ft, with
Notes
- ↑ Carleson & Gamelin 1993, pp. 28–32
- ↑ Shapiro 1993, pp. 90–93
References
- Berkson, E.; Porta, H. (1978), "Semigroups of analytic functions and composition operators", Michigan Math. J., 25: 101–115, doi:10.1307/mmj/1029002009
- Carleson, L.; Gamelin, T. D. W. (1993), Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-97942-5
- Elin, M.; Shoikhet, D. (2010), Linearization Models for Complex Dynamical Systems: Topics in Univalent Functions, Functional Equations and Semigroup Theory, Operator Theory: Advances and Applications, 208, Springer, ISBN 978-3034605083
- Koenigs, G.P.X. (1884), "Recherches sur les intégrales de certaines équations fonctionnelles", Ann. Sci. Ecole Norm. Sup., 1: 2–41
- Kuczma, Marek (1968). Functional equations in a single variable. Monografie Matematyczne. Warszawa: PWN – Polish Scientific Publishers. ASIN: B0006BTAC2
- Shapiro, J. H. (1993), Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, ISBN 0-387-94067-7
- Shoikhet, D. (2001), Semigroups in geometrical function theory, Kluwer Academic Publishers, ISBN 0-7923-7111-9