On a Bernoulli shift with non-identical factor measures
3.4. Recurrence
Pr o p o s itio n 2. Let T he the shift constructed in § 3.1. For an'/ measurable set E with PlE)>0, there exist for P-a.e. <o an infinite number of t such that T'ui e E for some i, tf,zi< m„ and that for such i, either
exp dr iiPTi — - (w) < (A, )‘ lM' -1 exp (»1,. I) dr holds.
Proof. We define a set B, by
B, ■ Jwe Í1: 1 /,+ 2p,<íl»-^,.|+w -N,♦2 + • • • + +
(27 By (11) we have
P(B,)>\p,.
Since the set B, is ^ r meMuratlle' ,he ,et* B\,Bt,... are independen with respect to the measure P. It follows from the Borel-Cantelli lemma (in th< independent case) that for P-a.e. u there exists an infinite number of t such (ha
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Bernoulli shift with twn-tdenlical factor measures 2S3
for all i6 Z and P-a.c. w. that is. the functions [JPT'vldP\iu>\, i s Z, have a uniformly positive lower hound and a uniform upper bound, where 7> is the induced transformation on the set E of T. However, this contradicts proposi
tion 2. —
Acknowledgement. The author would like to thank W . Krieger for calling his attention to U . Krengcl's paper [3], and M . Osikawa and Y. Ito for valuable discussions. He also would like to thank the Institut für Angewandte Mathematik. Universität Heidelberg, and Le Centre de Physique Théorique, CNRS, Marseille, for their hospitality while part of this work was done. The research was supported in part by the Alexander von Humboldt Stiftung.
R E FE R E N C E S
[I
] R. V, Chacon At I). S. Ornstcin A cenemi ergodie theorem Illinois J. Math. 4 (19601, l.'.t-I* 1'. [2| S Knkulani. On equivalence of infinite product measures. Ann. Math. 49 ( 194MI, 214-214. [J] U. Krcngel Transformation» without Unite invariant measure have timlc strong generator«.- 175 -
REFERENCES.
A: An zal, H .: On an example o f a measure preserving transformation which 1s not conjugate to I t s Inverse. Proc. Jap. Acad. 27 (1951), 517-522.
B: B illin g s le y , P .: Ergodlc theory and Inform ation. W iley, New York, 1965.
BAL: Bowen, R ., and Lanford, O .E .: Zeta functions o f re s tric tio n s o f the s h if t transform ation. Proc. Symp. Pure Math., Vol. 14 (A.M.S. 1970), 43-49.
Bol: Bowen, R .: P e rio d ic points and measures fo r Axiom A d1ffeomorph1sms. Trans. A.M.S. 154 (1971), 377-397.
Bo2: Bowen, R .: Smooth p a rtitio n s o f Anosov diffeomorphlsms are weak B e rn o u lli. Isra e l J . Math. 21 (1975), 95-100.
Bo3: Bowen, R .: Topological entropy and Axiom A. Proc. Symp. Pure Math., V o l. 14 (A.M.S. 1970), 23-41.
Bo4: Bowen, R .: Equ ilibrium States and the Ergodlc Theory o f Anosov Diffeomorphlsms. Lecture Notes 1n Mathematics, 470 (1975). S p rln ge r-V e rla g, Berl1n-He1delberg-New York.
Bu: B u tle r, R .: M.Sc. D iss e rta tio n , Warwick, 1978.
BAS: B u tle r, R ., and Schmidt, K .: An Information cocycle fo r groups o f non-singular transform ations. P re p rin t, U n ive rsity of Warwick, 1982. CAF: Chacon, R .V ., and Frlednan, N .: Approximation and In varia n t
measures. Z e lt. Wahr. Geb. 3 (1965), 286-295.
Ch: Choksl, J .R .: Extension o f a theorem o f E. Hopf. Journal Lond. Math. Soc. 36 (1961), 81-88. Corrigendum: 36 (1961), 253.
- 176 -
C&K: Cuntz, J . , and Krleger, W.: A class o f C -algebras and to p o logica l Harkov chains. In ven t, math. 56 (1980), 251-268.
D: Doob, J . L . : Stochastic processes. W ile y, New York, 1953. Du: Dugundjl, J . : Topology. A lly n and Bacon, Boston, 1966. Dy: Dye, H .A .: On groups o f measure preserving transformations I .
Amer. J . Math. 81 (1959), 119-159.
FAM: Feldnan, J . , and Moore, C .C .: Ergodic equivalence re la tio n s , cohomology and von Neumann algebras I . Trans. A.M.S. 234 (1977), 289-324.
F: F e lle r , W.: An Introduction to p ro b a b ility theory and Its a p p lic a tio n s. Vols. I and I I . W ile y, New York, 1957 and 1966. Fe: F e llg e t t , R .: Ph.D. Thesis, Warwick, 1976.
F&P: F e llg e t t , R ., and P arry, W.: Endomorphlsms of a Lebesgn space I I . B u ll. Lond. Math. S oc., 7 (1975), 151-158.
F r: Friechian, N .: Introduction to ergodic theory. Van Nostrand
Re Inhold Studies No. 29, New York, 1970. ,
HI: Halmos, P .R .: Lectures on ergodic theory. Chelsea, New York, 1956. H2: Halmos, P .R .: Measure theory. Van Nostrand, Princeton, N .J ., 1950. HAN: Halmos, P .R ., and von Neumann, J . : Operator methods 1n classical
mechanics I I . Annals o f Mathematics 43 (1942), 332-350. Ha: Hamachl, T . : On a Bernoulli s h if t w ith non-1dent1cal fa cto r
measures. Ergod. Th. & Oynam. Sys. 1 (1981), 273-283. I : Ionescu Tulcea, A .: On the category o f certain classes o f
transformations 1n ergodic theory. Trans. A.M.S. 114 (1965), 261-279.
JKKMAS: del Junco, A ., Keane, M., Kitchens, B ., Markus, B ., and Swanson, L .: Continuous homomorphlsms o f B ernoulli s h ifts . Proceedings of the Special Year 1n Ergodic Theory and Dynamical Systems, 1979-1980, U n iv e rs ity of Maryland. Progress 1n Mathematics 10, pages 91-111, Blrkhauser, S tu ttg a rt, 1981.
- 177 -
J&R: del Junco, A ., and Rahe, M.: F1n1tary codings and weak Bernoulli p a r titio n s . Proc. A.M.S. 75 (1979), 259-264.
K: Kakutani, S .: On equivalence of I n f in it e product measures. Ann. o f H ath., 49 (1948), 214-224.
Kr: Krengel, U .: Transformations without f in i t e In v a ria n t measure have f in it e strong generators. Lecture Notes 1n Mathematics, 160 (1970), 133-157. S p rln ge r-V e rla g, Berl1n-He1delberg-New York.
Kr1l: K rieger, W.: On non-singular transformations o f a measure space I . Z. Wahrsche1n11chke1tstheor1e verw. Geb. 11 (1969), 98-119. Kr12: K rieg e r, W.: On dimension functions and topological Markov chains.
Invent, math. 56 (1980), 239-250.
K&S: Krzyzewski, K., and Szlenk, W.: On In va ria n t measures fo r expanding d iffe re n tia b le mappings. Studla Mathematlca 33 (1969), 83-92.
LAY: Lasota, A . , and Yorke, J .A .: On the existence of In va ria n t measures f o r piecewise monotonic transform ations. Trans. A.M.S. 186 (1973), 481-488.
I MAE: M artin, N .F .G ., and England, J .H .: Mathematical Theory o f Entropy,
Encyclopedia o f Mathematics and It s A p p lic a tio n s ., V o l. 12, Addlson-Wesley, Reading, 1981.
N: Neveu, J . : Discrete-Parameter M artingales. North-Holland P ub lishing C o ., Amsterdam, 1975.
P I: P a rry , W.: Entropy and generators 1n ergodlc theory. Benjamin, New York, 1969.
P2: P a rry, H .: F1n1tary Isomorphisms with f in it e expected code-lengths. B u ll. Lond. Math. Soc. 11 (1979), 170-176.
P3: P a rry, W.: The Information cocycle and c-bounded codes. Israe l J . Math. 29 (1978), 205-220.
- 178 -
P4: P arry, W.: Topics 1n ergodlc th eory. Canbridge U n ive rsity Press, Cambridge, 1981.
P5: P arry, W.: An ergodlc theorem o f Information theory without In va ria n t measure. Proc. Lond. Hath. Soc. (3) 13 (1963), 605-612. P6: P arry, W .: Endomorphlsms of a Lebesgue space I I I . Israe l J . Math.
21 (1975), 167-72.
PAS: P arry, W ., and Schmidt, K.: P re p rin t, to appear.
PAT1: P arry, W ., and Tuncel, S .: On the stochastic and topological
structure o f Markov chains. B u ll. Lond. Math. Soc. 14 (1982), 16-27. PAT2: P arry, W ., and Tuncel, S .: C la s s ific a tio n Problems 1n Ergodlc Theory.
Lond. Math. Soc. Lecture Note S eries, 67. Cambridge U n iv e rsity Press, Cambridge, 1982.
PAT3: P arry, W ., and Tuncel, S .: On the c la s s ific a tio n o f Markov chains by f in it e equivalence. Ergod. Th. & Dynam. Sys. 1 (1981), 303-335. PAW: P arry, W ., and W illia m s, R .F .: Block coding and a zeta function
fo r f i n i t e Markov chains. Proc. Lond. Math. Soc. 35 (1977), 483-495. Pa: Parthasarathy, K .R .: P ro b a b ility measures on metric spaces.
Academic Press, New Y ork-London, 1967.
P1: Plnsker, M .S.: Information and Information s t a b ili t y o f random variables and processes. Izd a t. Akad. Nauk. SSSR., 1960 (Russian) ■ Holden-Day, San Francisco, 1964.
PuASh: Pugh, C . , and Shub, M.: E rg o d ld ty o f Anosov action s. Inventlones math. 15 (1972), 1-23.
R I: RohHn, V .A .: Lectures on the entropy theory o f measure-preserving transformations. Russian Math. Surveys 22 (1967), 1-52.
- 178 -
P4: P a rry, W.: Topics In ergodlc th eory. Cambridge U n iv e rsity Press, Cambridge, 1981.
P5: P a rry , W.: An ergodlc theorem o f Information theory w ithout In v a ria n t measure. Proc. Lond. Math. Soc. (3) 13 (1963), 605-612. P6: P a rry, W.: Endomorphlsms o f a Lebesgue space I I I . Israe l J . Math.
21 (1975), 167-72.
PAS: P a rry, W., and Schmidt, K.: P re p rin t, to appear.
PAT1: P a rry, W., and Tuncel, $ .: On the stochastic and topological
structure o f Markov chains. B u ll. Lond. Math. Soc. 14 (1982), 16-27. PAT2: P a rry , W., and Tuncel, S .: C la s s ific a tio n Problems 1n Ergodlc Theory.
Lond. Math. Soc. Lecture Note S eries, 67. Cambridge U n iv e rs ity Press, Cambridge, 1982.
PAT3: P a rry, W., and Tuncel, S .: On the c la s s ific a tio n o f Markov chains by f in it e equivalence. Ergod. Th. A Dynam. Sys. 1 (1981), 303-335. PAW: P a rry , W., and W illiam s, R .F .: Block coding and a zeta function
fo r f in it e Markov chains. P roc. Lond. Math. Soc. 35 (1977), 483-495. Pa: Parthasarathy, K .R .: P ro b a b ility measures on metric spaces.
Academic Press, New York-Lo ndo n, 1967.
P1: Plnsker, M .S.: Information and Information s t a b ilit y o f random variables and processes. Iz d a t. Akad. Nauk. SSSR., 1960 (Russian) ■ Holden-Day, San Francisco, 1964.
PuASh: Pugh, C ., and Shub, M.: E rg o d ld ty o f Anosov actions. Inventlones math. 15 (1972), 1-23.
R I: RohUn, V .A .: Lectures on the entropy theory o f measure-preserving transformations. Russian Math. Surveys 22 (1967), 1-52.
- 179 -
R2: RohUn, V .A .: Selected topics from the metric theory o f dynamical systems. Uspekhl Hat. Nauk, 4 (1949), 57-128 ■ A.M.S. Translations (2) 49 (1966), 171-240.
Ru: R u e lle , D .: Thermodynamic Formalism. Encyclopedia o f Mathematics and It s A p p lic a tio n s ,V o l. 5, Addison-Wesley, Reading, Mass. (1978).
S: Sacksteder, R .: The measures In v a ria n t under an expanding map. Lecture Notes 1n Mathematics, 392 (1974), 179-194. Sprlnger-Verlag, Berl1n-He1delberg-New York.
Sc: Schmidt, K.: Cocycles of ergodlc transformation groups. MacMillan ( In d ia ) , 1977.
Sel: Seneta, E .: Non-negative M atrices. A lle n and Unwin, London, 1973. Se2: Seneta, E .: Non-negative Matrices and Markov chains. 2 . Ed..
Springer Series 1n S ta tis tic s , Sp rln ger-V erlag, New York-Heldelberg- B e rlln , 1981.
Sh: Shub, M.: Endomorphlsms o f compact d iffe re n tia b le manifolds. . Amer. J . Math. 91 (1969), 175-199.
T1: Tuncel, S .: Conditional pressure and coding. Isra e l J . Math. 39 (1981), 101-112.
T2: Tuncel, S .: M.Sc. D isse rta tio n , Warwick, 1979.
Wl: W alters, P .: Ergodlc Theory - Introductory Lectures. Lecture Notes 1n Mathematics, 458 (1975). Sp rln ge r-V e rla g, B e rlln-H e lde lb erg- New York.
W2: W alters, P .: In va ria n t measures and equilibrium states fo r some mappings which expand distances. Trans. A.M.S. 236 (1978), 121-153.