Disintegration of projective measures
Authors:
Dorin Ervin Dutkay and Palle E. T. Jorgensen
Journal:
Proc. Amer. Math. Soc. 135 (2007), 169179
MSC (2000):
Primary 42C40, 42A16, 42A65, 43A65, 46G15, 47D07, 60G18
Published electronically:
June 22, 2006
MathSciNet review:
2280185
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper, we study a class of quasiinvariant 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]
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Anantharaman and Jean
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and physics (Boulder, CO, 1999) Contemp. Math., vol. 282, Amer.
Math. Soc., Providence, RI, 2001, pp. 35–46. MR 1855241
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Ola
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E. T. Jorgensen, and Vasyl′
Ostrovs′kyĭ, Representation theory and numerical
AFinvariants. The representations and centralizers of certain states on
𝒪_{𝒹}, Mem. Amer. Math. Soc. 168
(2004), no. 797, xviii+178. MR 2030387
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Ola
Bratteli and Palle
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Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA,
2002. The world of the spectrum. MR 1913212
(2003i:42001)
 [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
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Ingrid
Daubechies, Ten lectures on wavelets, CBMSNSF Regional
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D.E. Dutkay; P.E.T. Jorgensen, Martingales, endomorphisms, and covariant systems of operators in Hilbert space, preprint 2004, arxiv math.CA/0407330
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Dorin
Ervin Dutkay and Palle
E. T. Jorgensen, Hilbert spaces of martingales
supporting certain substitutiondynamical systems, Conform. Geom. Dyn. 9 (2005), 24–45
(electronic). MR
2133804 (2006a:37005), http://dx.doi.org/10.1090/S1088417305001359
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Freire, Artur
Lopes, and Ricardo
Mañé, An invariant measure for rational maps,
Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 45–62.
MR 736568
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Alex Kumjian; Jean Renault, KMS states on algebras associated to expansive maps, Preprint: arXiv/math.OA/0305044.
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no. 1, 27–43. MR 736567
(85m:58110a), http://dx.doi.org/10.1007/BF02584743
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(82h:46075)
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Jean
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pp. 371–386. MR 1770333
(2001g:46130)
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Jean
Renault, AF equivalence relations and their cocycles, Operator
algebras and mathematical physics (Constanţa, 2001) Theta,
Bucharest, 2003, pp. 365–377. MR 2018241
(2005e:46129)
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David
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Peter
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a function satisfying Bowen’s condition, Trans. Amer. Math. Soc. 353 (2001), no. 1, 327–347 (electronic). MR 1783787
(2001g:37029), http://dx.doi.org/10.1090/S0002994700026568
 [AnRe01]
 Claire Anantharaman; Jean Renault, Amenable groupoids. Groupoids in analysis, geometry, and physics (Boulder, CO, 1999), 3546, Contemp. Math., 282, Amer. Math. Soc., Providence, RI, 2001. MR 1855241 (2002g:46110)
 [BJO04]
 Ola Bratteli; Palle E. T. Jorgensen; Vasyl Ostrovskyi, Representation theory and numerical AFinvariants. The representations and centralizers of certain states on , Mem. Amer. Math. Soc. 168 (2004), no. 797, xviii+178 pp. MR 2030387 (2005i:46069)
 [BrJo02]
 O. Bratteli; P.E.T. Jorgensen, Wavelets through a looking glass: The world of the spectrum, Appl. Numerical and Harmonic Anal., Birkhauser, Boston, 2002. MR 1913212 (2003i:42001)
 [BrJo05]
 Ola Bratteli; Palle E. T. Jorgensen, Preprint in preparation.
 [Bro]
 H. Brolin, Invariant sets under iteration of rational functions, Arkiv för Matematik, 6, 1965, 103144. MR 0194595 (33:2805)
 [Dau92]
 I. Daubechies, Ten lectures on wavelets, CBMSNSF, Regional Conf. Ser. SIAM, vol. 61, 1992. MR 1162107 (93e:42045)
 [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]
 D.E. Dutkay; P.E.T. Jorgensen, Hilbert spaces of martingales supporting certain substitutiondynamical systems. Conform. Geom. Dyn. 9 (2005), 2445. MR 2133804 (2006a:37005)
 [FLM]
 A. Freire; A. Lopes; R. Mane, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 4562. MR 0736568 (85m:58110b)
 [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), 18981901. MR 1067632 (92a:81068)
 [Mane]
 R. Mane, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Bras. Mat. 14, 1983, 2743. MR 0736567 (85m:58110a)
 [Ren80]
 Jean Renault, A groupoid approach to algebras, Lecture Notes in Mathematics, 793. Springer, Berlin, 1980. ii+160 pp. MR 0584266 (82h:46075)
 [Ren98]
 Jean Renault, Cuntzlike algebras, Operator theoretical methods (Timisoara, 1998), 371386, Theta Found., Bucharest, 2000. MR 1770333 (2001g:46130)
 [Ren03]
 Jean Renault, AF equivalence relations and their cocycles, Operator algebras and mathematical physics (Constanta, 2001), 365377, Theta, Bucharest, 2003. MR 2018241 (2005e:46129)
 [Rue89]
 D. Ruelle, The thermodynamic formalism for expansive maps, Comm. Math. Phys. 125 (1989), 239262. MR 1016871 (91a:58149)
 [Wal01]
 Peter Walters, Convergence of the Ruelle operator for a function satisfying Bowen's condition, Trans. Amer. Math. Soc. 353 (2001), 327347. MR 1783787 (2001g:37029)
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Additional Information
Dorin Ervin Dutkay
Affiliation:
Department of Mathematics, Hill CenterBusch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 088548019
Email:
ddutkay@math.rutgers.edu
Palle E. T. Jorgensen
Affiliation:
Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, Iowa 522421419
Email:
jorgen@math.uiowa.edu
DOI:
http://dx.doi.org/10.1090/S0002993906084693
PII:
S 00029939(06)084693
Keywords:
Measures,
projective limits,
transfer operator,
martingale,
fixedpoint
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
