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Non-ergodicity for C1 expanding maps and g-measures

Published online by Cambridge University Press:  19 September 2008

Anthony N. Quasf
Anthony N. Quasf
Affiliation:
Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB, England
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Abstract

We introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 3 , June 1996 , pp. 531 - 543
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[1]Berbee, H.. Chains with infinite connections: uniqueness and Markov representation. Probab. Th. Rel. Fields 76 (1987), 243–253.CrossRefGoogle Scholar
[2]Bose, C. J.. Generalized baker's transformations. Ergod. Th. & Dynam. Sys. 9 (1989), 1–17.CrossRefGoogle Scholar
[3]Boyarsky, A. and Byers, W.. A graph-theoretic bound on the number of independent absolutely continuous invariant measures. J. Math. Anal. Appl. 139 (1989), 139–151.CrossRefGoogle Scholar
[4]Boyarsky, A. and Scarowsky, M.. On a class of transformations which have unique absolutely continuous invariant measure. Trans. Amer. Math. Soc. 255 (1979), 243–262.CrossRefGoogle Scholar
[5]Bramson, M. and Kalikow, S. A.. Nonuniqueness in g-functions. Israel J. Math. 84 (1993), 153–160.CrossRefGoogle Scholar
[6]Doeblin, W. and Fortet, R.. Sur les chaînes a liaisons complétes. Bull. Soc. Math. France 65 (1937), 132–148.Google Scholar
[7]Góra, P. and Schmitt, B.. Un exemple de transformation dilatante et C 1 par morceaux de l'intervalle, sans probabilité absolument continue invariante. Ergod. Th. & Dynam. Sys. 9 (1989), 101–113.CrossRefGoogle Scholar
[8]Hulse, P.. Uniqueness and ergodic properties of ergodic g-measures. Ergod. Th. & Dynam. Sys. 11 (1991), 65–77.CrossRefGoogle Scholar
]Kaijser, T.. On a theorem of Karlin. Preprint. Department of Electrical Engineering, Linköping University, Sweden, 1993.Google Scholar
[10]Kalikow, S. A.. Random Markov processes and uniform martingales. Israel J. Math. 71 (1990), 33–54.CrossRefGoogle Scholar
[11]Karlin, S.. Some random walks arising in learning models, I. Pac. J. Math. 3 (1953), 725–756.CrossRefGoogle Scholar
[12]Keane, M.. Strongly mixing g-measures. Invent. Math. 16 (1972), 309–324.CrossRefGoogle Scholar
[13]Kowalski, Z. S.. Invariant measures for piecewise monotonic transformations. Proc. Winter School of Probability (Karpacz, 1975). Springer Lecture Notes in Mathematics 472 (1975), 77–94.CrossRefGoogle Scholar
[14]Krzyzewski, K.. On expanding mappings. Bull. Acad. Polon. Sci. Sér. Sci. Math. XIX (1971), 23–24.Google Scholar
[15]Krzyzewski, K.. A remark on expanding mappings. Colloq. Math. XLI (1979), 291–295.CrossRefGoogle Scholar
[16]Krzyzewski, K. and Szlenk, W.. On invariant measures for expanding differentiable mappings. Studia Math. XXXIII (1969), 83–92.CrossRefGoogle Scholar
[17]Lasola, A. and Yorke, J. A.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.Google Scholar
[18]Li, T. Y. and Yorke, J. A.. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc. 235 (1978), 183–192.Google Scholar
[19]Lindvall, T.. Lectures on the Coupling Method. Wiley, New York, 1992.Google Scholar
[20]Mañé, R.. Ergodic Theory and Differentiable Dynamics. Springer, New York, 1988.Google Scholar
[21]Quas, A. N.. Some problems in ergodic theory. Thesis, University of Warwick, 1993.Google Scholar
[22]Walters, P.. Ruelle's operator theorem and g-measures. Trans. Amer. Math. Soc. 214 (1975), 375–387.Google Scholar
[23]Wong, S.. Some metric properties of piecewise monotonic mappings of the unit interval. Trans. Amer. Math. Soc. 246 (1978), 493–500.Google Scholar