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- Title
Analyticity of Smooth Eigenfunctions and Spectral Analysis of the Gauss Map.
- Authors
Antoniou, I.; Shkarin, S. A.
- Abstract
We provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form Uf(x)=∫a(x,y)f(b(x,y))μ(dy) acting on functions <MATH>f: [u, v]\rightarrow {\Bbb C}</MATH> (evolution operators of one-dimensional dynamical systems and Markov processes have this form). We estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the Frobenius–Perron operator of the continuous fraction Gauss map. We prove that any infinitely differentiable eigenfunction f of this Frobenius–Perron operator, corresponding to a non-zero eigenvalue admits a (unique) analytic extension to the set <MATH>{\Bbb C}\setminus (-\infty , -1]</MATH>. Analyzing the spectrum of the Frobenius–Perron operator in spaces of smooth functions, we extend significantly the domain of validity of the Mayer and Röpstorff asymptotic formula for the decay of correlations of the Gauss map.
- Subjects
SPIN eigenfunctions; EIGENFUNCTIONS; NUMERICAL solutions to boundary value problems; GAUSS maps; MATHEMATICS; MARKOV Processes International LLC
- Publication
Journal of Statistical Physics, 2003, Vol 111, Issue 1/2, p355
- ISSN
0022-4715
- Publication type
Article
- DOI
10.1023/A:1022217410549