TY - JOUR
T1 - On semiconjugation of entire functions
AU - Bergweiler, Walter
AU - Hinkkanen, A.
PY - 1999/5
Y1 - 1999/5
N2 - Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with g ○ f = h ○ g. We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g-1(J(h)) = J(f). Here J(·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h, there are only countably many entire functions g such that g ○ f - h ○ g.
AB - Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with g ○ f = h ○ g. We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g-1(J(h)) = J(f). Here J(·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h, there are only countably many entire functions g such that g ○ f - h ○ g.
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U2 - 10.1017/S0305004198003387
DO - 10.1017/S0305004198003387
M3 - Article
AN - SCOPUS:22644448778
SN - 0305-0041
VL - 126
SP - 565
EP - 574
JO - Mathematical Proceedings of the Cambridge Philosophical Society
JF - Mathematical Proceedings of the Cambridge Philosophical Society
IS - 3
ER -