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Ergodic properties of the Erdös measure, the entropy of the goldenshift, and related problems

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Abstract

We define a two-sided analog of the Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on\(\mathbb{T}^2 \) that is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.

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References

  1. Abramov LM (1959) The entropy of an induced automorphism. Sov Dokl128: 647–650

    Google Scholar 

  2. Adler RL, Weiss B (1967) Entropy, a complete metric invariant for automorphisms of the torus. Proc Nat Acad Sci USA57: 1573–1576

    Google Scholar 

  3. Alexander JC, Yorke JA (1984) Fat baker's transformations. Ergod Theory Dynam Systems4: 1–23

    Google Scholar 

  4. Alexander JC, Zagier D (1991) The entropy of a certain infinitely convolved Bernoulli measure. J London Math Soc44: 121–134

    Google Scholar 

  5. Avez A (1972) Entropie des groupes de type fini. CR Acad Sci Paris275A: 1363–1366

    Google Scholar 

  6. Belinskaya RM (1970) Generalized powers of an automorphism and entropy. Siberian Math J11: 739–749

    Google Scholar 

  7. Bertand-Mathis A (1986) Developpement en base θ, répartition modulo un de la suite (xθ n)n⩾0; langages codés et θ-shift. Bull Math Soc Fr114: 271–323

    Google Scholar 

  8. Carlitz L (1968) Fibonacci representations 'I. Fibonacci Quart6: 193–220

    Google Scholar 

  9. Cassels J (1957) An Introduction in Diophantine Approximation. Cambridge: Univ Press

    Google Scholar 

  10. Dajani K, Kraaikamp C, Solomyak B (1996) The natural extension of the β-transformation. Acta Math Hung73: 97–109

    Article  Google Scholar 

  11. Derrenic Y (1980) Quelques applications du théorème ergodique sous-additif. Astérisque74: 183–201

    Google Scholar 

  12. Dumont JM, Sidorov N, Thomas A (1998) Number of representations related to a linear recurrent basis. Preprint of Institut de Mathématiques de Luminy, Marseille, Prétirage n96-26

  13. Erdös P (1939) On a family of symmetric Bernoulli convolutions. Amer J Math61: 974–976

    Google Scholar 

  14. Erdös P, Joó I, Komornik V (1990) Characterization of the unique expansions\(1 = \sum _{i = 1}^\infty q^{ - n_i } \) and related problems. Bull Soc Math Fr118: 377–390

    Google Scholar 

  15. Frougny Ch (1992) Representations of numbers and finite automata. Math Systems Theory25: 37–60

    Article  Google Scholar 

  16. Frougny Ch,Sakarovitch J, Automatic conversion from Fibonacci to golden mean, and generalization. Int J Algebra and Comput (to appear)

  17. Garsia A (1963) Entropy and singularity of infinite convolutions. Pac J Math13: 1159–1169

    Google Scholar 

  18. Gelfond AO (1959) On a certain general property of number systems. Izvestiya Akad Nauk SSSR, Ser math23: 809–814

    Google Scholar 

  19. Jessen B, Wintner A (1938) Distribution functions and the Riemann zeta function. Trans Amer Math Soc38: 48–88

    Google Scholar 

  20. Kaimanovich V, Vershik A (1983) Random walks on discrete group: boundary and entropy. Ann Prob11: 457–490

    Google Scholar 

  21. Kenyon R, Vershik A (1998) Arithmetic construction of sofic partitions of hyperbolic toral automorphisms. Erg Theory Dyn Sys18: 357–372

    Article  Google Scholar 

  22. Lagarias JC (1992) Number theory and dynamical systems. Proc Symp Applied Math46: 35–72

    Google Scholar 

  23. Lalley S (1998) Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution. J London Math Soc (to appear)

  24. Le Borgne S (1997) Dynamique symbolique et propriétés stochastiques des automorphisms du tore: cas hyperbolique et quasi-hyperbolique. Thèse doctorale

  25. Ledrappier F, Porzio A (1994) A dimension formula for Bernoulli convolutions. J Stat Phys76: 1307–1327

    Google Scholar 

  26. Le Veque WJ (1956) Topics in Number Theory. Addison-Wesley

  27. Livshits AN, Vershik AM (1992) Adic models of ergodic transformations, spectral theory and related topics. Adv in Soviet Math9: 185–204

    Google Scholar 

  28. Ornstein D (1974) Ergodic Theory, Randomness and Dynamical Systems. New Haven: Yale Univ Press

    Google Scholar 

  29. Parry W (1960) On the β-expansions of real numbers. Acta Math Hungar11: 401–416

    Google Scholar 

  30. Pushkarev I (1995) Multizigzag ideal lattices and the enumeration of Fibonacci partitions. Zap Nauchn Sem POMI223: 280–312 (in Russian)

    Google Scholar 

  31. Rényi A (1957) Representations for real numbers and their ergodic properties. Acta Math Hungar8: 477–493

    Google Scholar 

  32. Sidorov NA (1995) The summation function for the number of Fibonacci representations. PDMI preprint 15

  33. Strâtilâ S, Voiculescu D (1975) Representations of AF algebras and of the groupU(∞). Lect Notes Math486, Berlin Heidelberg New York: Springer

    Google Scholar 

  34. Vershik AM (1981) Uniform algebraic approximation of shift and multiplication operators. Dokl Akad Nauk SSSR259: 526–529; English transl. Soviet Math Dokl24: 97–100

    Google Scholar 

  35. Vershik AM (1982) A theorem on Markov periodic approximation in ergodic theory. Zap Nauchn Sem LOMI115: 72–82 (in Russian); English transl. J Soviet Math28: 667–673 (1985)

    Google Scholar 

  36. Vershik AM (1994) Locally transversal symbolic dynamics. Algebra and Analysis6: (in Russian); English transl. St Petersburg Math J6: 529–540 (1995)

  37. Vershik AM (1977) Multivalued mappings with invariant measure (polymorphisms) and Markov operators. Zap Nauchn Sem LOMI72: 26–61 (in Russian); English transl. J Soviet Math23: 2243–2266 (1983)

    Google Scholar 

  38. Vershik AM (1991/1992) The fibadic expansions of real numbers and adic transformation. Prep Report Inst Mittag-Leffler, pp 1–9

  39. Vershik AM (1992) Arithmetic isomorphism of the toral hyperbolic automorphisms and sofic systems. Funct Anal Appl26: 22–24

    Article  Google Scholar 

  40. Vershik A, Kerov S (1985) Locally semisimple algebras. Combinatorial theory andK 0-functor. Current problems in Mathematics. Newest results. Itogi Nauki i Tehniki. VINITI26: 3–56; English transl. J Soviet Math38: 1701–1733 (1987)

    Google Scholar 

  41. Vershik A, Sidorov N (1993) Arithmetic expansions associated with rotation of the circle and with continued fractions. Algebra and Analysis5: 97–115 (in Russian); English transl. St Petersburg Math J5: 1121–1136 (1994)

    Google Scholar 

  42. Young L-S (1982) Dimension, entropy and Lyapunov exponents. Ergod Theory Dynam Systems2: 109–124

    Google Scholar 

  43. Sidorov N, Vershik A (1998) Bijective arithmetic codings of hyperbolic automorphisms of the 2-torus, and binary quadratic forms. J Dynam Control Systems4: 365–400

    Article  Google Scholar 

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Authors and Affiliations

  1. St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 19011, St. Petersburg, Russia

    Nikita Sidorov & Anatoly Vershik

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  1. Nikita Sidorov
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  2. Anatoly Vershik
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Additional information

With 11 Pigures

To the memory of Paul Erdös

Supported in part by the INTAS grant 93-0570. The first author was supported by the French foundation PRO MATHEMATICA. The first author expresses his gratitude to l'Institut de Mathématiques de Luminy for support during his stay in Marseille in 1996-97. The second author is grateful to the University of Stony Brook for support during his visit in February–March 1996 and to the Institute for Advanced studies of Hebrew University for support during his being there in 1997

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Sidorov, N., Vershik, A. Ergodic properties of the Erdös measure, the entropy of the goldenshift, and related problems. Monatshefte für Mathematik 126, 215–261 (1998). https://doi.org/10.1007/BF01367764

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