The Equilibrium States for Semigroups of Rational Maps
نویسندگان
چکیده
In this paper, we frequently use the notation from [S1]. A “rational semigroup” G is a semigroup generated by non-constant rational maps g : CI → CI, where CI denotes the Riemann sphere, with the semigroup operation being functional composition. For a rational semigroup G, we set F (G) := {z ∈ CI | G is normal in a neighborhood of z} and J(G) := CI \ F (G). F (G) is called the Fatou set of G and J(G) is called the Julia set of G. If G is generated by a family {fi}i, then we write G = 〈f1, f2, . . . 〉. The research on the dynamics of rational semigroup was initiated by Hinkkanen and Martin ([HM]), who were interested in the role of the dynamics of polynomial semigroups while studying various one-complex-dimensional moduli spaces for discrete groups, and by F. Ren’s group ([ZR]), who studied such semigroups from the perspective of random complex dynamics. The theory of the dynamics of rational semigroups is deeply related to that of the fractal geometry. In fact, If G = 〈f1, . . . , fs〉 is a finitely generated rational semigroup, then the Julia set J(G) of G has the “backward self-similarity”, i.e.,
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تاریخ انتشار 2006