Abstract
LetR be an expanding rational function with a real bounded Julia set, and let\(\left( {Lg} \right)\left( x \right) = \sum\limits_{Ry = x} {\frac{{g\left( y \right)}}{{\left[ {R'\left( y \right)} \right]^2 }}} \) be a Ruelle operator acting in a space of functions analytic in a neighbourhood of the Julia set. We obtain explicit expressions for the resolvent function\(E\left( {x,z;\lambda } \right) = \left( {I - \lambda L} \right)^{ - 1} \frac{1}{{z - x}}\) and, in particular, for the Fredholm determinantD(λ)=det(I-λL). It gives us an equation for calculating the escape rate. We relate our results to orthogonal polynomials with respect to the balanced measure ofR. Two examples are considered.
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Communicated by J.-P. Eckmann
The first named author was sponsored in part by the Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany)
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Levin, G.M., Sodin, M.L. & Yuditski, P.M. A Ruelle operator for a real Julia set. Commun.Math. Phys. 141, 119–132 (1991). https://doi.org/10.1007/BF02100007
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DOI: https://doi.org/10.1007/BF02100007