Abstract
We study a class of dynamical systems in \(L^2\) spaces of infinite products \(X\). Fix a compact Hausdorff space \(B\). Our setting encompasses such cases when the dynamics on \(X = B^\mathbb {N}\) is determined by the one-sided shift in \(X\), and by a given transition-operator \(R\). Our results apply to any positive operator \(R\) in \(C(B)\) such that \(R1 = 1\). From this we obtain induced measures \(\Sigma \) on \(X\), and we study spectral theory in the associated \(L^2(X,\Sigma )\). For the second class of dynamics, we introduce a fixed endomorphism \(r\) in the base space \(B\), and specialize to the induced solenoid \(\mathrm{Sol }(r)\). The solenoid \(\mathrm{Sol }(r)\) is then naturally embedded in \(X = B^\mathbb {N}\), and \(r\) induces an automorphism in \(\mathrm{Sol }(r)\). The induced systems will then live in \(L^2(\mathrm{Sol }(r), \Sigma )\). The applications include wavelet analysis, both in the classical setting of \(\mathbb {R}^n\), and Cantor-wavelets in the setting of fractals induced by affine iterated function systems (IFS). But our solenoid analysis includes such hyperbolic systems as the Smale-Williams attractor, with the endomorphism \(r\) there prescribed to preserve a foliation by meridional disks. And our setting includes the study of Julia set-attractors in complex dynamics.
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Acknowledgments
One of the authors wishes to thank Professors Ka-Sing Lau, De-Jun Feng, and their colleagues, for organizing a wonderful conference in Hong-Kong, “The International Conference on Advances of Fractals and Related Topics”, December 2012. Many discussions with participants at the conference inspired this paper. This work was partially supported by a grant from the Simons Foundation (#228539 to Dorin Dutkay).
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Dutkay, D.E., Jorgensen, P.E.T. (2014). The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative. In: Feng, DJ., Lau, KS. (eds) Geometry and Analysis of Fractals. Springer Proceedings in Mathematics & Statistics, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43920-3_3
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DOI: https://doi.org/10.1007/978-3-662-43920-3_3
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