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Real analyticity of Jacobian of invariant measures for hyperbolic meromorphic functions
Author(s) -
Badeńska Agnieszka
Publication year - 2008
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdn083
Subject(s) - mathematics , meromorphic function , jacobian matrix and determinant , invariant (physics) , bounded function , neighbourhood (mathematics) , analytic function , schwarzian derivative , conformal map , pure mathematics , probability measure , mathematical analysis , mathematical physics
We prove that for a hyperbolic meromorphic function f having a rapid derivative growth, if HD ( J ( f ))>1, then the Jacobian D μ ϕ of a probability invariant measure μ ϕ on J ( f ), equivalent to a conformal measure m ϕ , has a real analytic extension on a neighbourhood of J ( f )∖ f −1 (∞) in ℂ. If, in addition, f satisfies a balanced derivative growth condition with constant exponents, then this extension is bounded in a neighbourhood of every pole of f .