spectral theorem
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2022 ◽  
Vol 6 (1) ◽  
pp. 93-100
Author(s):  
Nırmal SARKAR ◽  
Sahın Injamamul ISLAM ◽  
Ashoke DAS

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 220
Author(s):  
Ezgi Erdoğan ◽  
Enrique A. Sánchez Pérez

A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.


Author(s):  
Hyungryul Baik ◽  
Inhyeok Choi ◽  
Dongryul M Kim

Abstract In this paper, we develop a way to extract information about a random walk associated with a typical Thurston’s construction. We first observe that a typical Thurston’s construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston’s construction that have finite 2nd moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss a random analogy of Kojima and McShane’s estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov monodromy. As another application, we discuss non-probabilistic estimations of stretch factors from Thurston’s construction and the powers for Salem numbers to become the stretch factors of pseudo-Anosovs from Thurston’s construction.


2021 ◽  
Vol 11 (1) ◽  
pp. 369-384
Author(s):  
Jifeng Chu ◽  
Fang-Fang Liao ◽  
Stefan Siegmund ◽  
Yonghui Xia ◽  
Hailong Zhu

Abstract For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.


2021 ◽  
pp. 193-208
Author(s):  
Rita Fioresi ◽  
Marta Morigi
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