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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2019 • Question 7

Question

Math 6210 • Real Analysis Spring 2019 • Question 7

Show that if \(f:\mathbb D\to \mathbb D\) is holomorphic then there exist \(n_1,n_2,\ldots \) so that \(f^{n_i}(z)\) converges pointwise for all \(z\in \mathbb D\). Note that \(f^{n_i}\) denotes \[ \underbrace {f\circ \cdots \circ f}_{n_i\ \text {times}}. \]

Raw LaTeX

Show that if \(f:\mathbb D\to\mathbb D\) is holomorphic then there
exist \(n_1,n_2,\ldots\) so that \(f^{n_i}(z)\) converges pointwise for all
\(z\in\mathbb D\). Note that \(f^{n_i}\) denotes
\[
\underbrace{f\circ\cdots\circ f}_{n_i\ \text{times}}.
\]

Topic Tags

Almost Everywhere and Convergence in Measure Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem