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arXiv · 1607.07229

Dynamical properties of endomorphisms, multiresolutions, similarity-and orthogonality relations

Palle Jorgensen·Feng Tian
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Abstract

We study positive transfer operators $R$ in the setting of general measure spaces $\left(X,\mathscr{B}\right)$. For each $R$, we compute associated path-space probability spaces $\left(\Omega,\mathbb{P}\right)$. When the transfer operator $R$ is compatible with an endomorphism in $\left(X,\mathscr{B}\right)$, we get associated multiresolutions for the Hilbert spaces $L^{2}\left(\Omega,\mathbb{P}\right)$ where the path-space $\Omega$ may then be taken to be a solenoid. Our multiresolutions include both orthogonality relations and self-similarity algorithms for standard wavelets and for generalized wavelet-resolutions. Applications are given to topological dynamics, ergodic theory, and spectral theory, in general; to iterated function systems (IFSs), and to Markov chains in particular.

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Palle Jorgensen, Feng Tian. 2016-07-25. Dynamical properties of endomorphisms, multiresolutions, similarity-and orthogonality relations. https://arxiv.org/abs/1607.07229

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