Tokyo Journal of Mathematics

On the Existence of a Darling-Kac Set for the Renormalized Rauzy Map

Kae INOUE and Hitoshi NAKADA

Full-text: Open access

Abstract

It is well-known that the renormalized Rauzy map is conservative and ergodic. In this paper, we show that a Darling-Kac set exists for the renormalized Rauzy map. This implies the pointwise dual ergodicity of this map.

Article information

Source
Tokyo J. Math., Volume 34, Number 2 (2011), 289-302.

Dates
First available in Project Euclid: 30 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1327931385

Digital Object Identifier
doi:10.3836/tjm/1327931385

Mathematical Reviews number (MathSciNet)
MR2918905

Zentralblatt MATH identifier
1268.37042

Citation

INOUE, Kae; NAKADA, Hitoshi. On the Existence of a Darling-Kac Set for the Renormalized Rauzy Map. Tokyo J. Math. 34 (2011), no. 2, 289--302. doi:10.3836/tjm/1327931385. https://projecteuclid.org/euclid.tjm/1327931385


Export citation

References

  • J. Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monograph 50, Amer Math Soc, Providence 1997.
  • A. Avila, and A. Bufetov, Exponential decay of correlations for the Rauzy-Veech-Zorich induction map, Partially hyperbolic dynamics, laminations, and Teichmuller flows, 203–211, Fields Inst. Commun., 51 (2007).
  • M. Keane, Non-ergodic interval exchange transformations, Israel J. Math. 26 (1977), no. 2, 188–196.
  • T. Morita, Renormalized Rauzy inductions, Probability and number theory–Kanazawa 2005, 263–288, Adv. Stud. Pure Math., 49, Math. Soc. Japan, Tokyo, 2007.
  • T. Morita, Renormalized Rauzy-Veech-Zorich inductions, Spectral analysis in geometry and number theory, 135–151, Contemp. Math., 484, Amer. Math. Soc., Providence, RI, 2009.
  • H. Nakada, R. Natsui, On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi-Perron algorithm, Monatsh. Math. 138 (2003), no. 4, 267–288.
  • G. Rauzy, Echanges d'intervalles et transformations induites, (French) Acta Arith. 34, (1979), no. 4, 315–328.
  • F. Schweiger, Kuzmin's theorem revisited, Ergodic Theory and Dynam Systems, 20 (2000), 557–565.
  • W. A. Veech, Interval exchange transformations, J. Analyse Math., 33 (1978), 222–278.
  • W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), 201–242.
  • A. Zorich, Finite Gauss measure on the space of interval exchange transformations, Lyapunov exponents. Ann. Inst. Fourier (Grenoble) 46 (1996), no. 2, 325–370.,
  • P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge. Philos. Soc., 53 (1957), 568-575.