Spectral and dynamical results related to certain non-integer base expansions on the unit interval

2025/02/10 by Horia D. Cornean, Cornean, Horia D., Ira Herbst +3
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2502.06511

openalex publication_date 2025/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider certain non-integer base β-expansions of Parry's type and we study various properties of the transfer (Perron-Frobenius) operator P:L^p([0,1])↦ L^p([0,1]) with p≥ 1 and its associated composition (Koopman) operator, which are induced by a discrete dynamical system on the unit interval related to these β-expansions. We show that if f is Lipschitz, then the iterated sequence \P^N f\_{N≥ 1} converges exponentially fast (in the L¹ norm) to an invariant state corresponding to the eigenvalue 1 of P. This "attracting" eigenvalue is not isolated: for 1≤ p≤ 2 we show that the point spectrum of P also contains the whole open complex unit disk and we explicitly construct some corresponding eigenfunctions.

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