Bicritical rational maps with a common iterate. (English) Zbl 1542.37033
This paper mainly focuses on the following two results for bicritical rational maps of degree \(d\):
(1) Let \(f\) and \(g\) be distinct bicritical rational maps and suppose there exists \(k \in\mathbb{N}\) such that \(f^k=g^k\). Then \(f\) and \(g\) have the same critical points and critical values;
(2) Let \(f\) and \(g\) be distinct bicritical maps that are not power maps and of even degree \(d\). If there exists \(k\) such that \(f^k=g^k\), then \(f^2=g^2\).
Note that converse of the first theorem is not true. Additionally, the authors investigate and clarify the deck group of \(f^k\) for any bicritical map.
(1) Let \(f\) and \(g\) be distinct bicritical rational maps and suppose there exists \(k \in\mathbb{N}\) such that \(f^k=g^k\). Then \(f\) and \(g\) have the same critical points and critical values;
(2) Let \(f\) and \(g\) be distinct bicritical maps that are not power maps and of even degree \(d\). If there exists \(k\) such that \(f^k=g^k\), then \(f^2=g^2\).
Note that converse of the first theorem is not true. Additionally, the authors investigate and clarify the deck group of \(f^k\) for any bicritical map.
Reviewer: Tao Chen (New York)
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37F12 | Critical orbits for holomorphic dynamical systems |
| 39B12 | Iteration theory, iterative and composite equations |
