1. The resonance spectrum of the cusp map in the space of analytic functions
- Author
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Ioannis Antoniou, Stanislav Shkarin, and Evgeny Yarevsky
- Subjects
Physics ,Cusp (singularity) ,Operator (physics) ,Spectrum (functional analysis) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Hardy space ,Eigenfunction ,Nonlinear Sciences - Chaotic Dynamics ,Space (mathematics) ,Section (fiber bundle) ,symbols.namesake ,symbols ,Chaotic Dynamics (nlin.CD) ,Mathematical Physics ,Mathematical physics ,Analytic function - Abstract
We prove that the Frobenius--Perron operator $U$ of the cusp map $F:[-1,1]\to[-1,1]$, $F(x)=1-2\sqrt{|x|}$ (which is an approximation of the Poincar�� section of the Lorenz attractor) has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1. We also prove that for any $q\in(0,1)$ the spectrum of $U$ in the Hardy space in the disk $\{z\in\C:|z-q, Submitted to JMP; The description of the spectrum in some Hardy spaces is added
- Published
- 2002