Ruelle operator and transcendental entire maps

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2005, 12(4): 773-789. doi: 10.3934/dcds.2005.12.773

Ruelle operator and transcendental entire maps

Patricia Domínguez ^1, , Peter Makienko ^2, and Guillermo Sienra ^3,

1.	F.C. Físico-Matemáticas, B.U.A.P, Av. San Claudio, Col. San Manuel, C.U., Puebla Pue., C.P. 72570, Mexico
2.	Instituto de Matemáticas, Unidad Cuernavaca. UNAM, Av. Universidad s/n. Col. Lomas de Chamilpa, C.P. 62210, Cuernavaca, Morelos, Mexico
3.	Facultad de Ciencias, UNAM, Av. Universidad 30, C.U., México D.F., C.P. 04510, Mexico

Received May 2003 Revised September 2004 Published January 2005

Abstract
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We calculate the Ruelle operator of a transcendental entire function $f$ having only a finite set of algebraic singularities. Moreover, under certain topological conditions on the postcritical set we prove (i) if $f$ has a summable critical point, then $f$ is not structurally stable and (ii) if all critical points of $f$ belonging to Julia set are summable, then there do not exist invariant lines fields on the Julia set.

Keywords: entire functions, Ruelle operator, Fatou set, Julia set, invariant line fields..

Mathematics Subject Classification: 37F10, 37F4.

Citation: Patricia Domínguez, Peter Makienko, Guillermo Sienra. Ruelle operator and transcendental entire maps. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 773-789. doi: 10.3934/dcds.2005.12.773

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