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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

Hilbert spaces of martingales supporting certain substitution-dynamical systems


Authors: Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal: Conform. Geom. Dyn. 9 (2005), 24-45
MSC (2000): Primary 42C40, 42A16, 42A65, 43A65, 46G15, 47D07, 60G18
Published electronically: March 24, 2005
MathSciNet review: 2133804
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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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Additional Information

Dorin Ervin Dutkay
Affiliation: Department of Mathematics Hill Center-Busch Campus, Rutgers University, Piscataway, New Jersey 08845-8019
Email: ddutkay@math.rutgers.edu

Palle E. T. Jorgensen
Affiliation: Department of Mathematics The University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242-1419
Email: jorgen@math.uiowa.edu

DOI: http://dx.doi.org/10.1090/S1088-4173-05-00135-9
PII: S 1088-4173(05)00135-9
Keywords: Measures, projective limits, transfer operator, martingale, fixed point
Received by editor(s): January 23, 2005
Received by editor(s) in revised form: February 2, 2005
Published electronically: March 24, 2005
Dedicated: Dedicated to the memory of Shizuo Kakutani
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.