Inverse limits, entropy and weak isomorphism for discrete dynamical systems

Author:
James R. Brown

Journal:
Trans. Amer. Math. Soc. **164** (1972), 55-66

MSC:
Primary 28A65

DOI:
https://doi.org/10.1090/S0002-9947-1972-0296251-7

MathSciNet review:
0296251

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Abstract: A categorical approach is taken to the study of a single measure-preserving transformation of a finite measure space and to inverse systems and inverse limits of such transformations. The questions of existence and uniqueness of inverse limits are settled. Sinai's theorem on generators is recast and slightly extended to say that entropy respects inverse limits, and various known results about entropy are obtained as immediate corollaries, e.g. systems with quasi-discrete or quasi-periodic spectrum have zero entropy. The inverse limit of an inverse system of dynamical systems is (1) ergodic, (2) weakly mixing, (3) mixing (of any order) iff each has the same property. Finally, inverse limits are used to lift a weak isomorphism of dynamical systems and to an isomorphism of systems and with the same entropy.

**[1]**L. M. Abramov,*Metric automorphisms with quasi-discrete spectrum*, Izv. Akad. Nauk SSSR Ser. Mat.**26**(1962), 513-530; English transl., Amer. Math. Soc. Transl. (2)**39**(1964), 37-56. MR**26**#606. MR**0143040 (26:606)****[2]**P. Billingsley,*Ergodic theory and information*, Wiley, New York, 1965. MR**33**#254. MR**0192027 (33:254)****[3]**J. R. Brown,*A universal model for dynamical systems with quasi-discrete spectrum*, Bull. Amer. Math. Soc.**75**(1969), 1028-1030. MR**39**#5770. MR**0244456 (39:5770)****[4]**J. R. Choksi,*Inverse limits of measure spaces*, Proc. London Math. Soc. (3)**8**(1958), 321-342. MR**20**#3251. MR**0096768 (20:3251)****[5]**H. Furstenberg,*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1-49. MR**35**#4369. MR**0213508 (35:4369)****[6]**P. R. Halmos,*Lectures on ergodic theory*, Publ. Math. Soc. Japan, no. 3, Math. Soc. Japan, Tokyo, 1956. MR**20**#3958. MR**0097489 (20:3958)****[7]**-,*Measure theory*, Van Nostrand, Princeton, N. J., 1950. MR**11**, 504. MR**0033869 (11:504d)****[8]**P. R. Halmos and J. von Neumann,*Operator methods in classical mechanics*. II, Ann. of Math. (2)**43**(1942), 332-350. MR**4**, 14. MR**0006617 (4:14e)****[9]**D. Maharam,*On homogeneous measure algebras*, Proc. Nat. Acad. Sci. U.S.A.**28**(1942), 108-111. MR**4**, 12. MR**0006595 (4:12a)****[10]**V. A. Rohlin,*Exact endomorphisms of a Lebesgue space*, Izv. Akad. Nauk SSSR Ser. Mat.**25**(1961), 499-530; English transl., Amer. Math. Soc. Transl. (2)**39**(1964), 1-36. MR**26**#1423. MR**0143873 (26:1423)****[11]**-,*Lectures on the entropy theory of transformations with invariant measure*, Uspehi Mat. Nauk**22**(1967), no. 5 (137), 3-56 = Russian Math. Surveys**22**(1967), no. 5, 1-52. MR**36**#349. MR**0217258 (36:349)****[12]**W. Rudin,*Fourier analysis on groups*, Interscience Tracts in Pure and Appl. Math., no. 12, Interscience, New York, 1962. MR**27**#2808. MR**0152834 (27:2808)****[13]**T. L. Seethoff,*Zero-entropy automorphisms of a compact abelian group*, Oregon State University, Dept. of Math. Technical Report #40, 1968.**[14]**J. G. Sinaĭ,*On a weak isomorphism of transformations with invariant measure*, Mat. Sb.**63**(**105**) (1964), 23-42; English transl., Amer. Math. Soc. Transl. (2)**57**(1966), 123-143. MR**28**#5164b; MR**28**, 1247. MR**0161961 (28:5164b)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0296251-7

Keywords:
Inverse limits,
dynamical systems,
measure-preserving transformation,
factor,
invariant subalgebra,
weakly isomorphic,
direct product,
bounded inverse system,
Lebesgue system,
discrete spectrum,
exact system,
natural extension,
disjoint,
ergodic,
weakly mixing,
mixing (of any order),
quasi-discrete spectrum,
quasi-periodic spectrum

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© Copyright 1972
American Mathematical Society