Abstract
In [20] the author and A. Vershik have shown that for β=1/2(1 + √5) and the alphabet {0,1} the infinite Bernoulli convolution (= the Erdős measure) has a property similar to the Lebesgue measure. Namely, it is quasi-invariant of type II1 under the β-shift, and the natural extension of the β-shift provided with the measure equivalent to the Erdős measure, is Bernoulli. In this note we extend this result to all Pisot parameters β (modulo some general arithmetic conjecture) and an arbitrary “sufficient” alphabet.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Sh. Akiyama, On the boundary of self-affine tiling generated by Pisot numbers, J. Math. Soc. Japan, 54 (2002), 283-308.
J. C. Alexander and D. Zagier, The entropy of a certain infinitely convolved Bernoulli measure, J. London Math. Soc., 44 (1991), 121-134.
D. Berend and Ch. Frougny, Computability by finite automata and Pisot bases, Math. Systems Theory, 27 (1994), 275-282.
K. Dajani, C. Kraaikamp and B. Solomyak, The natural extension of the β-transformation, Acta Math. Hungar., 73 (1996), 97-109.
P. Erdős, On a family of symmetric Bernoulli convolutions, Amer. J. Math., 61 (1939), 974-975.
Ch. Frougny, Representations of numbers and finite automata, Math. Systems Theory, 25 (1992), 37-60.
Ch. Frougny and B. Solomyak, Finite beta-expansions, Ergodic Theory Dynam. Systems, 12 (1992), 713-723.
A. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc., 102 (1962), 409-432.
M. Hollander, Linear Numeration Systems, Finite Beta Expansions, and Discrete Spectrum of Substitution Dynamical Systems, Ph.D. Thesis, University of Washington, 1996.
E. Olivier, N. Sidorov and A. Thomas, On the Gibbs properties of Bernoulli convolutions related to β-numeration, preprint.
D. Ornstein, Ergodic Theory, Randomness and Dynamical Systems, Yale Univ. Press (New Haven and London, 1974).
W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hung., 11 (1960), 401-416.
A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung., 8 (1957) 477-493.
K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc., 12 (1980), 269-278.
K. Schmidt, Algebraic codings of expansive group automorphisms and two-sided beta-shifts, Monatsh. Math., 129 (2000), 37-61.
N. Sidorov, Bijective and general arithmetic codings for Pisot toral automorphisms, J. Dynam. Control Systems, 7 (2001), 447-472.
N. Sidorov, An arithmetic group associated with a Pisot unit, and its symbolic-dynamical representation, Acta Arith., 101 (2002), 199-213.
N. Sidorov, Arithmetic Dynamics, in: Topics in Dynamics and Ergodic Theory, LMS Lecture Notes, 310, Cambridge Univ. Press (2003).
N. Sidorov, Almost every number has a continuum of β-expansions, to appear in Amer. Math. Monthly.
N. Sidorov and A. Vershik, Ergodic properties of Erdős measure, the entropy of the goldenshift, and related problems, Monatsh. Math., 126 (1998), 215-261.
M. Smorodinsky, β-automorphisms are Bernoulli shifts, Acta Math. Acad. Sci. Hung., 24 (1973), 273-278.
B. Solomyak, On the random series σ ± λ i (an Erdős problem), Annals of Math., 142 (1995), 611-625.
A. Vershik, Arithmetic isomorphism of the toral hyperbolic automorphisms and sofic systems, Functional. Anal. Appl., 26 (1992), 170-173.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Szidorov, N. Ergodic-theoretic properties of certain Bernoulli convolutions. Acta Mathematica Hungarica 101, 345–355 (2003). https://doi.org/10.1023/B:AMHU.0000004944.27485.2a
Issue Date:
DOI: https://doi.org/10.1023/B:AMHU.0000004944.27485.2a