Abstract
We show there exists a constant 0 < c0 < 1 such that the dimension of every measure on [0, 1], which makes the digits in the continued fraction expansion independent, is at most 1 − c0. This extends a result of Kifer, Peres and Weiss from 2001, which established this under the additional assumption of stationarity. For k ≥ 1 we prove an analogous statement for measures under which the digits form a *-mixing k-step Markov chain. This is also generalized to the case of f-expansions. In addition, we construct for each k a measure, which makes the continued fraction digits a stationary and *-mixing k-step Markov chain, with dimension at least 1 − 23−k.
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References
R. Adler, Continued fractions and Bernoulli trials, in Ergodic Theory, Courant Institute of Mathematical Sciences, New York University, New York, 1975, pp. 111–120.
N. Alon and J. Spencer, The Probabilistic Method, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1992.
P. Billingsley, Probability and Measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York 1995.
P. Billingsley and I. Henningsen, Hausdorff dimension of some continued-fraction sets, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 31 (1974/75), 163–173.
J. R. Blum, D. L. Hanson and L. H. Koopmans, On the strong law of large numbers for a class of stochastic processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 2 (1963), 1–11.
R. C. Bradley, On the ψ-mixing condition for stationary random sequences, Transactions of the American Mathematical Society 276 (1983), 55–66.
S. D. Chatterji, Masse, die von regelmässigen Kettenbrüchen induziert sind, Mathematische Annalen 164 (1966), 113–117.
M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, Vol. 259, Springer, London, 2011.
L. Heinrich, Mixing properties and central limit theorem for a class of non-identical piecewise monotonic C2-transformations, Mathematische Nachrichten 181 (1996), 185–214.
Y. Kifer, Y. Peres and B. Weiss, A dimension gap for continued fractions with independent digits, Israel Journal of Mathematics 124 (2001), 61–76.
J. R. Kinney and T. S. Pitcher, The dimension of some sets defined in terms of f-expansions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 4 (1966), 293–315
A. Rényi, Representations for real numbers and their ergodic properties, Acta Mathematica Academiae Scientiarum Hungaricae 8 (1957), 477–493.
P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Transactions of the American Mathematical Society 236 (1975), 121–153.
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer, New York–Berlin, 1982.
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Rapaport, A. A dimension gap for continued fractions with independent digits—the non stationary case. Isr. J. Math. 227, 911–930 (2018). https://doi.org/10.1007/s11856-018-1753-6
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DOI: https://doi.org/10.1007/s11856-018-1753-6