Abstract
This note contains a new proof of Host’s equidistribution theorem for multiplicatively independent endomorphisms of ℝ/ℤ. The method is a simplified version of our recent work on equidistribution under toral automorphisms [2] and is related to the argument in [3], but avoids the use of the scenery flow and of Marstrand’s projection theorem, using instead a direct Fourier argument to establish smoothness of the limit measure.
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References
W. Feller, An Introduction to Probability Theory and Its Applications. Vol. II, John Wiley & Sons, New York, 1971.
M. Hochman, Toral endomorphisms and equidistribution, preprint.
M. Hochman and P. Shmerkin, Equidistribution from fractal measures, Inventiones Mathematicae 202 (2015), 427–479.
B. Host, Nombres normaux, entropie, translations, Israel Journal of Mathematics 91 (1995), 419–428.
E. Lindenstrauss, p-adic foliation and equidistribution, Israel Journal of Mathematics 122 (2001), 29–42.
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To Benjy Weiss, who set me on my way
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Hochman, M. A short proof of Host’s equidistribution theorem. Isr. J. Math. 251, 527–539 (2022). https://doi.org/10.1007/s11856-022-2444-x
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DOI: https://doi.org/10.1007/s11856-022-2444-x