Abstract
We construct a planar version of the natural extension of the piecewise linear transformation T generating greedy β-expansions with digits in an arbitrary set of real numbers A = {a 0, a 1, a 2}. As a result, we derive in an easy way a closed formula for the density of the unique T-invariant measure µ absolutely continuous with respect to Lebesgue measure. Furthermore, we show that T is exact and weak Bernoulli with respect to µ.
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References
G. Brown and Q. Yin, β-transformation, natural extension and invariant measure, Ergodic Theory Dynam. Systems, 20 (2000), 1271–1285.
K. Dajani and C. Kalle, A note on the greedy β-transformation with arbitrary digits, Preprint 1362, Utrecht University, 2007.
K. Dajani and C. Kalle, Random β-expansions with deleted digits, Disc. and Cont. Dyn. Sys., 18 (2007), 199–217.
K. Dajani, C. Kraaikamp and B. Solomyak, The natural extension of the β-transformation, Acta Math. Hungar., 73 (1996), 97–109.
Karma Dajani and Martijn de Vries, Invariant densities for random β-expansions, J. Eur. Math. Soc. (JEMS), 9 (2007), 157–176.
P. Erdős, I. Joó and V. Komornik, Characterization of the unique expansions 1 = \( \sum\nolimits_{i = 1}^\infty {q^{ - n_i } } \) and related problems, Bull. Soc. Math. France, 118 (1990), 377–390.
A. O. Gel’fond, A common property of number systems, Izv. Akad. Nauk SSSR. Ser. Mat., 23 (1959), 809–814.
P. Góra, Invariant densities for piecewise linear and increasing maps of constant slope, http://www.mathstat.concordia.ca/faculty/pgora/deleted/Variable_signed_july14.pdf.
M. Keane, M. Smorodinsky and B. Solomyak, On the morphology of γ-expansions with deleted digits, Trans. Amer. Math. Soc., 347 (1995), 955–966.
C. Kopf, Invariant measures for piecewise linear transformations of the interval, Appl. Math. Comput., 39 (1990), 123–144.
W. Parry, On the β-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401–416.
M. Pedicini, Greedy expansions and sets with deleted digits, Theoret. Comput. Sci., 332 (2005), 313–336.
M. Pollicott and K. Simon, The Hausdorff dimension of λ-expansions with deleted digits, Trans. Amer. Math. Soc., 347 (1995), 967–983.
A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477–493.
V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499–530.
M. Rychlik, Bounded variation and invariant measure, Studia Math., 76 (1983), 69–80.
N. Sidorov, Almost every number has a continuum of β-expansions, Amer. Math. Monthly, 110 (2003), 838–842.
K. Wilkinson, Ergodic properties of a class of piecewise linear transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1974/75), 303–328.
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Dajani, K., Kalle, C. A natural extension for the greedy β-transformation with three arbitrary digits. Acta Math Hung 125, 21–45 (2009). https://doi.org/10.1007/s10474-009-8212-0
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DOI: https://doi.org/10.1007/s10474-009-8212-0