Hilbert spaces of martingales supporting certain substitution-dynamical systems

Dutkay, Dorin; Jorgensen, Palle

doi:10.1090/S1088-4173-05-00135-9

Hilbert spaces of martingales supporting certain substitution-dynamical systems
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by Dorin Ervin Dutkay and Palle E. T. Jorgensen

Conform. Geom. Dyn. 9 (2005), 24-45

DOI: https://doi.org/10.1090/S1088-4173-05-00135-9

Published electronically: March 24, 2005

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Abstract:

Let $X$ be a compact Hausdorff space. We study finite-to-one mappings $r\colon X\rightarrow X$, onto $X$, and measures on the corresponding projective limit space $X_\infty (r)$. We show that certain quasi-invariant measures on $X_\infty (r)$ correspond in a one-to-one fashion to measures on $X$ which satisfy two identities. Moreover, we identify those special measures on $X_\infty (r)$ which are associated via our correspondence with a function $V$ on $X$, a Ruelle transfer operator $R_V$, and an equilibrium measure $\mu _V$ on $X$.

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