×

Covering numbers: Arithmetics and dynamics for rotations and interval exchanges. (English) Zbl 0996.37006

Suppose that \(R_{\alpha}\) is an irrational rotation of the circle. Consider the symbolic expansion of \(R_{\alpha}\) with respect to the partition \(\{[0,\beta),[\beta ,1)\}\) (as usual the circle is identified with \([0,1)\)). Then the lengths of \(n\)-cylinders of \(R_{\alpha}\) are obviously the distances of points of the form \(k\alpha\) or \(\beta +l\alpha\) with \(0\leq k,l\leq n-1\), if \(n\) is large enough. According to the five distance theorem these distances can take at most five different values. The authors relate these distances with the continued fraction approximation of \(\alpha\). An algorithm is described, which describes the “best left approximations” of \(\beta\) by \(k\alpha\).
Then different kinds of covering numbers are considered. Again some of them are related with the continued fraction approximation of \(\alpha\).
Finally, an interval exchange transformation \(T\) of three intervals is considered. Results on covering numbers are obtained. Then the authors derive interesting results on \(T\). For example, if \(T\) is ergodic, then \(T\) has simple spectrum. Some ergodic properties of \(T\) are again related with the continued fraction approximation of certain parameters of \(T\).
Reviewer: Peter Raith (Wien)

MSC:

37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
28D05 Measure-preserving transformations
37E05 Dynamical systems involving maps of the interval
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] [ALE] P. Alessandri,Codages de rotations et basses complexités, Ph.D. Thesis, Université Aix-Marseille II, 1996.
[2] Alessandri, P.; Berthé, V., Three distance theorem and combinatorics on words, Enseign. Math., 44, 103-132 (1998) · Zbl 0997.11051
[3] Arnold, V. I., Small denominators and problems of stability of motion in classical and celestial mechanics, Uspeki Mat. Nauk, 18, 6, 91-192 (1963) · Zbl 0135.42701
[4] Arnoux, P.; Ferenczi, S.; Hubert, P., Trajectories of rotations, Acta. Arith., 87, 209-217 (1999) · Zbl 0921.11033
[5] Arnoux, P.; Rauzy, G., Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math. France, 119, 199-215 (1991) · Zbl 0789.28011
[6] Berstel, J., Recent results in Sturmian words, Developments in Language Theory II (Magedburg 1995), 13-24 (1996), Singapore: World Scientific, Singapore · Zbl 1096.68689
[7] Berthé, V., Fréquences des facteurs des suites sturmiennes, Theoret. Comput. Sci., 165, 295-309 (1996) · Zbl 0872.11018 · doi:10.1016/0304-3975(95)00224-3
[8] Boshernitzan, M., A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J., 52, 723-752 (1985) · Zbl 0602.28009 · doi:10.1215/S0012-7094-85-05238-X
[9] [BOS-NOG] M. Boshernitzan and A. Nogueira,Mixing properties of interval exchange transformations in preparation.
[10] [CHA] R. V. Chacon,A geometric construction of measure-preserving transformations, inProceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1965, pp. 335-360. · Zbl 0212.08401
[11] Chekhova, N., Covering numbers of rotations, Theoret. Comput. Sci., 230, 97-116 (1999) · Zbl 0947.68543 · doi:10.1016/S0304-3975(97)00256-9
[12] Coven, E. M.; Hedlund, G. A., Sequences with minimal block growth, Math. Systems Theory, 7, 138-153 (1973) · Zbl 0256.54028 · doi:10.1007/BF01762232
[13] del Junco, A., Transformations with discrete spectrum are stacking transformations, Canad. J. Math., 24, 836-839 (1976) · Zbl 0312.47003
[14] Ferenzci, S., Systèmes localement de rang un, Ann. Inst. H. Poincaré, 20, 35-51 (1984) · Zbl 0535.28010
[15] Ferenczi, S., Tiling and local rank properties of the Morse sequence, Theoret. Comput. Sci., 129, 369-383 (1994) · Zbl 0820.68092 · doi:10.1016/0304-3975(94)90034-5
[16] Ferenczi, S., Les transformations de Chacon: combinatoire, structure géométrique, lien avec les systèmes de complexité 2n+1, Bull. Soc. Math. France, 123, 271-292 (1995) · Zbl 0855.28008
[17] Ferenczi, S., Systems of finite rank, Colloq. Math., 73, 35-65 (1997) · Zbl 0883.28014
[18] Geelen, A. S.; Simpson, R. J., A two dimensional Steinhaus theorem, Australas. J. Combin., 8, 169-197 (1993) · Zbl 0804.11020
[19] [GOO] G. R. Goodson,Functional equations associated with the spectral properties of compact group extensions, inProceedings of Conference on Ergodic Theory and its Connections with Harmonic Analysis Alexandria 1993, Cambridge University Press, 1994, pp. 309-327. · Zbl 0947.28014
[20] Guenais, M., Une majoration de la multiplicité spectrale d’opérateurs associés à des cocycles réguliers, Israel J. Math., 105, 263-283 (1998) · Zbl 0911.28013 · doi:10.1007/BF02780334
[21] Hedlund, G. A.; Morse, M., Symbolic dynamics, Amer. J. Math., 60, 815-866 (1938) · Zbl 0019.33502 · doi:10.2307/2371264
[22] Hedlund, G. A.; Morse, M., Symbolic dynamics II. Sturmian trajectories, Amer. J. Math., 62, 1-42 (1940) · Zbl 0022.34003 · doi:10.2307/2371449
[23] Kalikow, S., Twofold mixing implies threefold mixing for rank-1 transformations, Ergodic Theory Dynam. Systems, 4, 237-259 (1984) · Zbl 0552.28016
[24] Katok, A. B.; Sataev, E. A., Standardness of automorphisms of transposition of intervals and fluxes on surfaces, Math. Zametki, 20, 4, 479-488 (1976) · Zbl 0368.58010
[25] Katok, A. B.; Stepin, A. M., Approximations in ergodic theory, Uspekhi Math. Nauk, 22, 5, 81-106 (1967) · Zbl 0172.07202
[26] Keane, M. S., Interval exchange transformations, Math. Z., 141, 25-31 (1975) · Zbl 0278.28010 · doi:10.1007/BF01236981
[27] Keynes, H.; Newton, D., A “minimal” non-uniquely ergodic interval exchange transformation, Math. Z., 148, 101-106 (1976) · Zbl 0308.28014 · doi:10.1007/BF01214699
[28] King, J. L., Joining-rank and the structure of finite-rank mixing transformations, J. Analyse Math., 51, 182-227 (1988) · Zbl 0665.28010
[29] Komatsu, T., On inhomogeneous continued fraction expansions and inhomogeneous diophantine approximation, J. Number Theory, 62, 192-212 (1997) · Zbl 0878.11029 · doi:10.1006/jnth.1997.2060
[30] Komatsu, T., The fractional part of nψ+ϕ and Beatty sequences, J. Théor. Nombres Bordeaux, 7, 387-406 (1995) · Zbl 0849.11027
[31] [LOT] M. Lothaire,Algebraic Combinatorics on Words, Chapter 2:Sturmian words, by J. Berstel and P. Séébold, to appear. · Zbl 1001.68093
[32] [ORN-RUD-WEI] D. S. Ornstein, D. J. Rudolph and B. Weiss,Equivalence of measure-preserving transformations, Mem. Amer. Math. Soc.262 (1982). · Zbl 0504.28019
[33] Oseledets, V. I., On the spectrum of ergodic automorphisms, Doklady Akad. Nauk SSSR, 168, 5, 1009-1011 (1966) · Zbl 0152.33404
[34] Ostrowski, A., Bemerkunger zur Theorie der Diophantischen Approximationen I, II, Abh. Math. Sem. Univ. Hamburg, I, 77-98 (1922) · JFM 48.0185.01
[35] [RAU1] G. Rauzy,Suites à termes dans un alphabet fini, Sém. Théor. Nombres Bordeaux (1983), 25-01-25-16. · Zbl 0547.10048
[36] Rauzy, G., Echanges d’intervalles et transformations induites, Acta Arith., 34, 315-328 (1979) · Zbl 0414.28018
[37] Rote, G., Sequences with subword complexity 2n., J. Number Theory, 46, 196-213 (1994) · Zbl 0804.11023 · doi:10.1006/jnth.1994.1012
[38] Slater, N. B., Gaps and steps for the sequence nϕ mod 1, Proc. Cambridge Phil. Soc., 63, 1115-1123 (1967) · Zbl 0178.04703
[39] Sós, V. T., On the distribution mod 1of the sequence nα, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 1, 127-134 (1958) · Zbl 0094.02903
[40] Sós, V. T., On the theory of diophantine approximations. I, Acta Math. Hungar., 8, 461-472 (1957) · Zbl 0080.03503 · doi:10.1007/BF02020329
[41] Sós, V. T., On the theory of diophantine approximations. II, Acta Math. Hungar., 9, 229-241 (1958) · Zbl 0086.03902 · doi:10.1007/BF02023874
[42] Sós, V. T., On a problem of Hartman about normal forms, Colloq. Math., 7, 155-160 (1960) · Zbl 0094.02805
[43] Surányi, J., Über die Anordnung der Vielfachen einer reellen Zahl mod 1, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 1, 107-111 (1958) · Zbl 0094.02904
[44] Świerczkowski, S., On successive settings of an arc on the circumference of a circle, Fund. Math., 46, 187-189 (1958) · Zbl 0085.27203
[45] Veech, W. A., Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2, Trans. Amer. Math. Soc., 140, 1-33 (1969) · Zbl 0201.05601 · doi:10.2307/1995120
[46] Veech, W. A., The metric theory of interval exchange transformations I, II, III, Amer. J. Math., 106, 1331-1421 (1984) · Zbl 0631.28006 · doi:10.2307/2374396
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.