Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Symmetric Gibbs measures


Authors: Karl Petersen and Klaus Schmidt
Journal: Trans. Amer. Math. Soc. 349 (1997), 2775-2811
MSC (1991): Primary 28D05, 60G09; Secondary 58F03, 60J05, 60K35, 82B05
DOI: https://doi.org/10.1090/S0002-9947-97-01934-X
MathSciNet review: 1422906
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 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 $C^*$-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 $[0,1]$, 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 $g$-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

Similar Articles

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

Retrieve articles in all journals with MSC (1991): 28D05, 60G09, 58F03, 60J05, 60K35, 82B05


Additional Information

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: https://doi.org/10.1090/S0002-9947-97-01934-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.
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society