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Entropy and uniform distribution of orbits in\(\mathbb{T}\)

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Abstract

We present a class of integer sequences {c n } with the property that for everyp-invariant and ergodic positive-entropy measure μ on L 2 \(\mathbb{T}\), {c n x (mod 1)} is uniformly distributed for μ-almost everyx. This extends a result of B. Host, who proved this for the sequence {q n}, forq relatively prime top. Our class of sequences includes, for instance, the sequencec n =Мf i (n)q ni , where the numbersq i are distinct and are relatively prime top andf i are any polynomials. More generally, recursion sequences for which the free coefficient of the recursion polynomial is relatively prime top are in this class as well, provided they satisfy a simple irreducibility condition.

In the multi-dimensional case we derive sufficient conditions for a pair of endomorphisms\(\mathbb{T}^d \) (withA diagonal) and anA-invariant and ergodic measure μ, such thatB-orbits of the form {B n ω} are uniformly distributed for μ-almost every\(\mathbb{T}^d \).

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  1. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel

    David Meiri

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  1. David Meiri
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Correspondence to David Meiri.

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Meiri, D. Entropy and uniform distribution of orbits in\(\mathbb{T}\) . Isr. J. Math. 105, 155–183 (1998). https://doi.org/10.1007/BF02780327

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  • DOI: https://doi.org/10.1007/BF02780327

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