Rigidity of escaping dynamics for transcendental entire functions. (English) Zbl 1226.37027
The author studies the dynamical behaviour near infinity of functions in the much studied Eremenko-Lyubich class \({\mathcal B}\), that is, the class of transcendental entire functions for which the set of singular values is bounded.
The main result of the paper is one of the major advances in the area of holomorphic dynamics in recent years. It states the following:
If \(f\) and \(g\) are two functions in \({\mathcal B}\) that are quasiconformally equivalent near infinity, then there exist \(R > 0\) and a quasiconformal map \(\varphi\) such that
\[ \varphi\circ f = g\circ\varphi \quad\text{on}\quad J_R(f) = \{z: |f^n(z)|\geq R\} \]
for all \(n\geq 1\). Furthermore, \(\varphi\) has zero dilatation on \(\{z\in J_R(f):|f^n(z)|\to\infty\}\). This explains the similarities that have been observed between the Julia sets of many entire functions.
The result can be seen as an analogue of a classical theorem of Böttcher which states that any two polynomials of the same degree \(d \geq 2\) are conformally conjugate near infinity.
The main result of the paper is one of the major advances in the area of holomorphic dynamics in recent years. It states the following:
If \(f\) and \(g\) are two functions in \({\mathcal B}\) that are quasiconformally equivalent near infinity, then there exist \(R > 0\) and a quasiconformal map \(\varphi\) such that
\[ \varphi\circ f = g\circ\varphi \quad\text{on}\quad J_R(f) = \{z: |f^n(z)|\geq R\} \]
for all \(n\geq 1\). Furthermore, \(\varphi\) has zero dilatation on \(\{z\in J_R(f):|f^n(z)|\to\infty\}\). This explains the similarities that have been observed between the Julia sets of many entire functions.
The result can be seen as an analogue of a classical theorem of Böttcher which states that any two polynomials of the same degree \(d \geq 2\) are conformally conjugate near infinity.
Reviewer: Liangwen Liao (Nanjing)
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
Keywords:
transcendental entire functions; singular values; quasiconformally equivalent; invariant line fieldsReferences:
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