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Register a new userEigenvalues of finite rank Bratteli-Vershik dynamical systems
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Authors
Xavier Bressaud
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In this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated to the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition to be a measurable eigenvalue. Then we consider two families of examples. A first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally we study Toeplitz type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove measurable eigenvalues are always rational but not necessarily continuous.
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In this article we characterize measure theoretical eigenvalues of Toeplitz Bratteli-Vershik minimal systems of finite topological rank which are not associated to a continuous eigenfunction. Several examples are provided to illustrate the different situations that can occur.
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