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 -