Transactions of the American Mathematical Society

Author(s): Karl Petersen; Klaus Schmidt
Journal: Trans. Amer. Math. Soc. 349 (1997), 2775-2811.
MSC (1991): Primary 28D05, 60G09; Secondary 58F03, 60J05, 60K35, 82B05
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Abstract: We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

1.: C. O. Acuna, Texture modeling using Gibbs distributions, Graphical Models and Image Processing 54 (1992), 210-222.
2.: D. J. Aldous, Exchangeability and related topics, École d'été de probabilités de Saint-Flour XIII-1983, Lecture Notes in Math., vol. 1117, Springer-Verlag, New York, 1985, pp. 2-198. MR 88d:60107
3.: L. K. Arnold, On $\sigma$ -finite invariant measures, Zeit. Wahr. verw. Geb. 9 (1968), 85-97. MR 37:2947
4.: D. Blackwell and D. Freedman, The tail $\sigma$ -field of a Markov chain and a theorem of Orey, Ann. Math. Stat. 35 (1964), 1291-1295. MR 29:1672
5.: R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, New York, 1975. MR 56:1364
6.: R. Butler and K. Schmidt, An information cocycle for groups of non-singular transformations, Zeit. Wahr. verw. Geb. 64 (1985), 347-360. MR 86i:28029
7.: P. Diaconis and D. Freedman, de Finetti's theorem for Markov chains, Ann. Prob. 8 (1980), 115-139. MR 81f:60090
8.: -, Partial exchangeability and sufficiency, Statistics: Applications and New Directions (Calcutta) (J. K. Ghosh and J. Roy, eds.), Indian Statistical Institute, Calcutta, 1984, pp. 205-236. MR 86i:60097
9.: E. B. Dynkin, Sufficient statistics and extreme points, Ann. Prob. 6 (1978), 705-730. MR 58:24575
10.: J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras. I Trans. Amer. Math. Soc. 234 (1977), 289-324. MR 58:28261a
11.: J. Feldman, C. Sutherland and R.J. Zimmer, Subrelations of ergodic equivalence relations, Erg. Th. Dyn. Sys. 9 (1989), 239-269. MR 91c:28020
12.: S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intelligence 6 (1984), 721-741.
13.: H.-O. Georgii, Canonical Gibbs states, their relation to Gibbs states, and applications to two-valued Markov chains, Zeit. Wahr. verw. Geb. 32 (1975), 277-300. MR 52:9417
14.: -, Canonical Gibbs Measures, Lecture Notes in Math, vol. 760, Springer-Verlag, New York, 1979. MR 81b:60108
15.: T. Giordano, I. Putnam and C. Skau, Topological orbit equivalence and -crossed products, J. Reine Angew. Math. 469 (1995), 51-111. CMP 96:5
16.: L. A. Grigorenko, On the $\sigma$ -algebra of symmetric events for a countable Markov chain, Theory Prob. Appls. 24 (1979), 199-204. MR 80d:60087
17.: A. Hajian, Y. Ito, and S. Kakutani, Invariant measures and orbits of dissipative transformations, Adv. in Math. 9 (1972), 52-65. MR 46:2003
18.: G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Sys. Th. 3 (1969), 320-375. MR 41:4510
19.: E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80 (1955), 470-501. MR 17:863g
20.: T. Höglund, Central limit theorems and statistical inference for finite Markov chains, Zeit. Wahr. verw. Geb. 29 (1974), 123-151. MR 51:9402
21.: R. Isaac, Generalized Hewitt-Savage theorems for strictly stationary processes, Proc. Amer. Math. Soc. 63 (1977), 313-316. MR 58:18695
22.: -, Note on a paper of J. L. Palacios, Proc. Amer. Math. Soc. 101 (1987), 529. MR 88k:60070
23.: S. Ito, A construction of transversal flows for maximal Markov automorphisms, Tokyo J. Math. 1 (1978), 305-324. MR 81c:28012
24.: S. Kakutani, A problem of equidistribution on the unit interval , Proceedings of the Oberwolfach Conference on Measure Theory (1975), Lecture Notes in Math., vol. 541, Springer-Verlag, New York, 1976, pp. 369-376. MR 56:15882
25.: M. Kowada, Spectral type of one-parameter group of unitary ooperators with transversal group, Nagoya Math.J. 32 (1968), 141-153. MR 37:5725
26.: -, The orbit-preserving transformation groups associated with a measurable flow, J. Math. Soc. Japan 24 (1972), 355-373. MR 46:5577
27.: W. Krieger, On the finitary isomorphisms of Markov shifts that have finite expected coding time, Zeit. Wahr. verw. Geb. 65 (1983), 323-328. MR 85e:28034
28.: I. Kubo, Quasi-flows, Nagoya Math. J. 35 (1969), 1-30. MR 40:301
29.: E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. für Math. 44 (1852), 115-116.
30.: S. L. Lauritzen, Extremal families and systems of sufficient statistics, Lecture Notes in Statistics, vol. 49, Springer-Verlag, New York, 1988. MR 90g:62010
31.: F. Ledrappier, Principe variationnel et systèmes dynamiques symboliques, Zeit. Wahr. verw. Geb. 30 (1974), 185-202. MR 53:8384
32.: G. Letta, Sur les théorèmes de Hewitt-Savage et de de Finetti, Seminaire de Probabilités XXIII, Lecture Notes in Math., vol. 1378, Springer-Verlag, New York, 1989, pp. 531-535. MR 91b:60030
33.: A. N. Livshitz, A sufficient condition for weak mixing of substitutions and stationary adic transformations, Math. Notes 44 (1988), 920-925. MR 90c:28027
34.: A. A. Lodkin and A. M. Vershik, Approximation for actions of amenable groups and transversal automrophisms, Lecture Notes in Math., vol. 1132, Springer-Verlag, New York, 1985, pp. 331-346. MR 87d:28017
35.: E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math. 1 (1878), 184-240.
36.: R. A. Olshen, The coincidence of measure algebras under an exchangeable probability, Zeit. Wahr. verw. Geb. 18 (1971), 153-158. MR 44:5992
37.: -, A note on exchangeable sequences, Zeit. Wahr. verw. Geb. 28 (1974), 317-321. MR 51:11637
38.: J.-L. Palacios, A correction note on ``Generalized Hewitt-Savage theorems for strictly stationary processes'', Proc. Amer. Math. Soc. 88 (1983), 138-140. MR 86a:60054
39.: -, The exchangeable sigma-field of Markov chains, Zeit. Wahr. verw. Geb. 69 (1985), 177-186. MR 86f:60082a
40.: W. Parry and S. Tuncel, Classification Problems in Ergodic Theory, LMS Lecture Note Series, vol. 67, Cambridge University Press, Cambridge, 1982. MR 84g:28024
41.: R. Pyke, The asymptotic behaviour of spacing under Kakutani's model for interval subdivision, Ann. Prob. 8 (1980), 157-163. MR 81c:60038
42.: P. Ressel, De Finetti-type theorems: An analytical approach, Ann. Prob. 13 (1985), 898-922. MR 86k:60023
43.: D. Ruelle, Statistical mechanics on a compact set with $\mathbb Z$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 185 (1973), 237-251. MR 54:5441
44.: C. Ryll-Nardzewski, Stationary sequences of random variables and the de Finetti's equivalence, Colloq. Math. 4 (1957), 149-156. MR 19:585e
45.: K. Schmidt, Cocycles on Ergodic Transformation Groups, MacMillan (India), Delhi, 1977. MR 58:28262
46.: -, Hyperbolic structure preserving isomorphisms of Markov shifts, Israel J. Math. 55 (1986), 213-228. MR 88c:28013
47.: -, Algebraic Ideas in Ergodic Theory, Conf. Board Math. Sci. Regional Conf. Ser. Math., vol. 76, American Mathematical Society, Providence, R.I., 1990. MR 92k:28029
48.: Ja. G. Sinai, Probabilistic ideas in ergodic theory, Amer. Math. Soc. Transls., Ser. 2, 31 (1963), 62-84. MR 32:196 (Russian original)
49.: E. Slud, A note on exchangeable sequences of events, Rocky Mountain J. Math. 8 (1978), 439-442. MR 57:17790
50.: B. Solomyak, On the spectral theory of adic transformations, Adv. Soviet Math. 9 (1992), 217-230. MR 93h:28028
51.: R. L. Thompson, Equilibrium states on thin energy shells, Mem. Amer. Math. Soc., No. 150, AMS, Providence, R.I., 1974. MR 51:12249
52.: A. M. Vershik, Description of invariant measures for actions of some infinite groups, Dokl. Akad. Nauk SSSR 218 (1974), 749-752 Soviet Math. Dokl. 15 (1974), 1396-1400. MR 51:8377
53.: -, Uniform algebraic approximation of shift and multiplication operators, Dokl. Akad. Nauk SSSR 259 (1981), 526-529=Soviet Math. Dokl. 24 (1981), 97-100. MR 83c:46064
54.: -, A theorem on the Markov periodic approximation in ergodic theory, J. Soviet Math. 28 (1985), 667-673. MR 84m:28030 (Russian original)
55.: A. M. Vershik and A. N. Livshitz, Adic models of ergodic transformations, spectral theory, substitutions, and related topics, Adv. Soviet Math. 9 (1992), 185-204. MR 93i:46131
56.: J. von Plato, The significance of the ergodic decomposition of stationary measures for the interpretation of probability, Synthese 53 (1982), 419-432. MR 84f:03021
57.: P. Walters, Ruelle's operator theorem and -measures, Trans. Amer. Math. Soc. 214 (1975), 375-387. MR 54:515
58.: R. J. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20 (1976), 373-409. MR 53:13522
59.: -, Cocycles and the structure of ergodic group actions, Israel J. Math. 26 (1977), 214-220. MR 55:10645

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 28D05, 60G09, 58F03, 60J05, 60K35, 82B05

Karl Petersen
Affiliation: Department of Mathematics, CB 3250, Phillips Hall, University of North Carolina, Chapel Hill, North Carolina 27599
Email: petersen@math.unc.edu

Klaus Schmidt
Affiliation: Department of Mathematics, University of Vienna, Vienna, Austria
Email: klaus.schmidt@univie.ac.at

DOI: 10.1090/S0002-9947-97-01934-X
PII: S 0002-9947(97)01934-X
Keywords: Gibbs measure, subshift of finite type, cocycle, Borel equivalence relation, exchangeability, adic transformation, tail field, interval splitting, Kolmogorov property, ratio limit theorem, Markov chain
Received by editor(s): August 17, 1995
Received by editor(s) in revised form: August 20, 1996
Additional Notes: First author supported in part by NSF Grant DMS-9203489.
Copyright of article: Copyright 1997, American Mathematical Society

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