Abstract
We study a particular case of the two-dimensional Steinhaus theorem, giving estimates of the possible distances between points of the formkα andkα+β on the unit circle, through an approximation algorithm of β by the pointskα. This allows us to compute covering numbers (maximal measures of Rokhlin stacks having certain prescribed regularity properties) for the symbolic dynamical systems associated to the rotation of argument α, acting on the partition of the circle by the points 0, β. We can the compute topological and measure-theoretic covering numbers for exchange of three intervals; in this way, we prove that every ergodic exchange of three intervals has simple spectrum and build a new class of three-interval exchanges which are not of rank one.
Similar content being viewed by others
References
[ALE] P. Alessandri,Codages de rotations et basses complexités, Ph.D. Thesis, Université Aix-Marseille II, 1996.
[ALE-BERT] P. Alessandri and V. Berthé,Three distance theorem and combinatorics on words, Enseign. Math.44 (1998), 103–132.
[ARNOL] V. I. Arnold,Small denominators and problems of stability of motion in classical and celestial mechanics, Uspeki Mat. Nauk18, 6 (1963), 91–192, translated in Russian Math. Surveys18, 6 (1963), 86–194.
[ARNOU-FER-HUB] P. Arnoux, S. Ferenczi and P. Hubert,Trajectories of rotations, Acta. Arith.87 (1999), 209–217.
[ARNOU-RAU] P. Arnoux and G Rauzy,Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math. France119 (1991), 199–215.
[BERS] J. Berstel,Recent results in Sturmian words, inDevelopments in Language Theory II (Magedburg 1995), World Scientific, Singapore, 1996, pp. 13–24.
[BERT] V. Berthé,Fréquences des facteurs des suites sturmiennes, Theoret. Comput. Sci.165 (1996), 295–309.
[BOS] M. Boshernitzan,A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J.52 (1985), 723–752.
[BOS-NOG] M. Boshernitzan and A. Nogueira,Mixing properties of interval exchange transformations in preparation.
[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.
[CHE] N. Chekhova,Covering numbers of rotations, Theoret. Comput. Sci.230 (1999), 97–116.
[COV-HED] E. M. Coven and G. A. Hedlund,Sequences with minimal block growth, Math. Systems Theory7 (1973), 138–153.
[deJ] A. del Junco,Transformations with discrete spectrum are stacking transformations, Canad. J. Math.24 (1976), 836–839.
[FER1] S. Ferenzci,Systèmes localement de rang un, Ann. Inst. H. Poincaré20 (1984), 35–51.
[FER2] S. Ferenczi,Tiling and local rank properties of the Morse sequence, Theoret. Comput. Sci.129 (1994), 369–383.
[FER3] S. Ferenczi,Les transformations de Chacon: combinatoire, structure géométrique, lien avec les systèmes de complexité 2n+1, Bull. Soc. Math. France123 (1995), 271–292.
[FER4] S. Ferenczi,Systems of finite rank, Colloq. Math.73 (1997), 35–65.
[GEE-SIM] A. S. Geelen and R. J. Simpson,A two dimensional Steinhaus theorem, Australas. J. Combin.8 (1993), 169–197.
[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.
[GUE] M. Guenais,Une majoration de la multiplicité spectrale d’opérateurs associés à des cocycles réguliers, Israel J. Math.105 (1998), 263–283.
[HED-MOR1] G. A. Hedlund and M. Morse,Symbolic dynamics, Amer. J. Math.60 (1938), 815–866.
[HED-MOR2] G. A. Hedlund and M. Morse,Symbolic dynamics II. Sturmian trajectories, Amer. J. Math.62 (1940), 1–42.
[KAL] S. Kalikow,Twofold mixing implies threefold mixing for rank-1 transformations, Ergodic Theory Dynam. Systems4 (1984), 237–259.
[KAT-SAT] A. B. Katok and E. A. Sataev,Standardness of automorphisms of transposition of intervals and fluxes on surfaces, Math. Zametki20, 4 (1976), 479–488 (in Russian), translated in Math. Notes Acad. Sci. USSR20, 4 (1977), 826–831.
[KAT-STE] A. B. Katok and A. M. Stepin,Approximations in ergodic theory, Uspekhi Math. Nauk22, 5 (1967), 81–106 (in Russian), translated in Russian Math. Surveys22, 5 (1967), 76–102.
[KEA] M. S. Keane,Interval exchange transformations Math. Z.141 (1975), 25–31.
[KEY-NEW] H. Keynes and D. Newton,A “minimal” non-uniquely ergodic interval exchange transformation. Math. Z.148 (1976), 101–106.
[KIN] J. L. King,Joining-rank and the structure of finite-rank mixing transformations, J. Analyse Math.51 (1988), 182–227.
[KOM1] T. Komatsu,On inhomogeneous continued fraction expansions and inhomogeneous diophantine approximation, J. Number Theory62 (1997), 192–212.
[KOM2] T. Komatsu,The fractional part of nψ+ϕ and Beatty sequences, J. Théor. Nombres Bordeaux7 (1995), 387–406.
[LOT] M. Lothaire,Algebraic Combinatorics on Words, Chapter 2:Sturmian words, by J. Berstel and P. Séébold, to appear.
[ORN-RUD-WEI] D. S. Ornstein, D. J. Rudolph and B. Weiss,Equivalence of measure-preserving transformations, Mem. Amer. Math. Soc.262 (1982).
[OSE] V. I. Oseledets,On the spectrum of ergodic automorphisms, Doklady Akad. Nauk SSSR168, 5 (1966), 1009–1011 (in Russian), translated in Soviet Math. Doklady7 (1966), 776–779.
[OST] A. Ostrowski,Bemerkunger zur Theorie der Diophantischen Approximationen I, II, Abh. Math. Sem. Univ. HamburgI (1922), 77–98, 250–251.
[RAU1] G. Rauzy,Suites à termes dans un alphabet fini, Sém. Théor. Nombres Bordeaux (1983), 25-01-25-16.
[RAU2] G. Rauzy,Echanges d’intervalles et transformations induites, Acta Arith.34 (1979), 315–328.
[ROT] G. Rote,Sequences with subword complexity 2n., J. Number Theory 46 (1994), 196–213.
[SLA] N. B. Slater,Gaps and steps for the sequence nϕ mod 1, Proc. Cambridge Phil. Soc.63 (1967), 1115–1123.
[SOS1] V. T. Sós,On the distribution mod 1of the sequence nα. Ann. Univ. Sci. Budapest. Eötvös Sect. Math.1 (1958), 127–134.
[SOS2] V. T. Sós,On the theory of diophantine approximations. I, Acta Math. Hungar.8 (1957), 461–472.
[SOS3] V. T. Sós,On the theory of diophantine approximations. II, Acta Math. Hungar.9 (1958), 229–241.
[SOS4] V. T. Sós,On a problem of Hartman about normal forms, Colloq. Math.7 (1960), 155–160.
[SUR] J. Surányi,Über die Anordnung der Vielfachen einer reellen Zahl mod 1, Ann. Univ. Sci. Budapest. Eötvös Sect. Math.1 (1958), 107–111.
[SWI] S. Świerczkowski,On successive settings of an arc on the circumference of a circle, Fund. Math.46 (1958), 187–189.
[VEE1] W. A. Veech,Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem mod 2, Trans. Amer. Math. Soc.140 (1969), 1–33.
[VEE2] W. A. Veech,The metric theory of interval exchange transformations I, II, III, Amer. J. Math.106 (1984), 1331–1421.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Berthé, V., Chekhova, N. & Ferenczi, S. Covering numbers: Arithmetics and dynamics for rotations and interval exchanges. J. Anal. Math. 79, 1–31 (1999). https://doi.org/10.1007/BF02788235
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02788235