Disintegration of projective measures

Authors:
Dorin Ervin Dutkay and Palle E. T. Jorgensen

Journal:
Proc. Amer. Math. Soc. **135** (2007), 169-179

MSC (2000):
Primary 42C40, 42A16, 42A65, 43A65, 46G15, 47D07, 60G18

Published electronically:
June 22, 2006

MathSciNet review:
2280185

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study a class of quasi-invariant measures on paths generated by discrete dynamical systems. Our main result characterizes the subfamily of these measures which admit a certain disintegration. This is a disintegration with respect to a random walk Markov process which is indexed by the starting point of the paths. Our applications include wavelet constructions on Julia sets of rational maps on the Riemann sphere.

**[AnRe01]**Claire Anantharaman and Jean Renault,*Amenable groupoids*, Groupoids in analysis, geometry, and physics (Boulder, CO, 1999) Contemp. Math., vol. 282, Amer. Math. Soc., Providence, RI, 2001, pp. 35–46. MR**1855241**, 10.1090/conm/282/04677**[BJO04]**Ola Bratteli, Palle E. T. Jorgensen, and Vasyl′ Ostrovs′kyĭ,*Representation theory and numerical AF-invariants. The representations and centralizers of certain states on 𝒪_{𝒹}*, Mem. Amer. Math. Soc.**168**(2004), no. 797, xviii+178. MR**2030387**, 10.1090/memo/0797**[BrJo02]**Ola Bratteli and Palle Jorgensen,*Wavelets through a looking glass*, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2002. The world of the spectrum. MR**1913212****[BrJo05]**Ola Bratteli; Palle E. T. Jorgensen, Preprint in preparation.**[Bro]**Hans Brolin,*Invariant sets under iteration of rational functions*, Ark. Mat.**6**(1965), 103–144 (1965). MR**0194595****[Dau92]**Ingrid Daubechies,*Ten lectures on wavelets*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR**1162107****[DuJo04a]**D.E. Dutkay; P.E.T. Jorgensen,*Martingales, endomorphisms, and covariant systems of operators in Hilbert space*, preprint 2004, arxiv math.CA/0407330**[DuJo04b]**Dorin Ervin Dutkay and Palle E. T. Jorgensen,*Hilbert spaces of martingales supporting certain substitution-dynamical systems*, Conform. Geom. Dyn.**9**(2005), 24–45 (electronic). MR**2133804**, 10.1090/S1088-4173-05-00135-9**[FLM]**Alexandre Freire, Artur Lopes, and Ricardo Mañé,*An invariant measure for rational maps*, Bol. Soc. Brasil. Mat.**14**(1983), no. 1, 45–62. MR**736568**, 10.1007/BF02584744**[KuRe03]**Alex Kumjian; Jean Renault,*KMS states on -algebras associated to expansive maps*, Preprint: arXiv/math.OA/0305044.**[Law90]**Wayne M. Lawton,*Tight frames of compactly supported affine wavelets*, J. Math. Phys.**31**(1990), no. 8, 1898–1901. MR**1067632**, 10.1063/1.528688**[Mane]**Ricardo Mañé,*On the uniqueness of the maximizing measure for rational maps*, Bol. Soc. Brasil. Mat.**14**(1983), no. 1, 27–43. MR**736567**, 10.1007/BF02584743**[Ren80]**Jean Renault,*A groupoid approach to 𝐶*-algebras*, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR**584266****[Ren98]**Jean Renault,*Cuntz-like algebras*, Operator theoretical methods (Timişoara, 1998) Theta Found., Bucharest, 2000, pp. 371–386. MR**1770333****[Ren03]**Jean Renault,*AF equivalence relations and their cocycles*, Operator algebras and mathematical physics (Constanţa, 2001) Theta, Bucharest, 2003, pp. 365–377. MR**2018241****[Rue89]**David Ruelle,*The thermodynamic formalism for expanding maps*, Comm. Math. Phys.**125**(1989), no. 2, 239–262. MR**1016871****[Wal01]**Peter Walters,*Convergence of the Ruelle operator for a function satisfying Bowen’s condition*, Trans. Amer. Math. Soc.**353**(2001), no. 1, 327–347 (electronic). MR**1783787**, 10.1090/S0002-9947-00-02656-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
42C40,
42A16,
42A65,
43A65,
46G15,
47D07,
60G18

Retrieve articles in all journals with MSC (2000): 42C40, 42A16, 42A65, 43A65, 46G15, 47D07, 60G18

Additional Information

**Dorin Ervin Dutkay**

Affiliation:
Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019

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/S0002-9939-06-08469-3

Keywords:
Measures,
projective limits,
transfer operator,
martingale,
fixed-point

Received by editor(s):
August 16, 2004

Received by editor(s) in revised form:
July 29, 2005

Published electronically:
June 22, 2006

Additional Notes:
This work was supported in part by NSF grant DMS 0457491

Communicated by:
David R. Larson

Article copyright:
© Copyright 2006
American Mathematical Society