### Current browse context:

math.DS

### Change to browse by:

### References & Citations

# Mathematics > Dynamical Systems

# Title: Equidistribution results for self-similar measures

(Submitted on 26 Feb 2020 (v1), last revised 22 Feb 2021 (this version, v3))

Abstract: A well known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In this paper we give sufficient conditions for an analogue of this theorem to hold for self-similar measures. Our approach applies more generally to sequences of the form $(f_{n}(x))_{n=1}^{\infty}$ where $(f_n)_{n=1}^{\infty}$ is a sequence of sufficiently smooth real valued functions satisfying a nonlinearity assumption. As a corollary of our main result, we show that if $C$ is equal to the middle third Cantor set and $t\geq 1$, then with respect to the Cantor-Lebesgue measure on $C+t$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed for almost every $x$.

## Submission history

From: Simon Baker [view email]**[v1]**Wed, 26 Feb 2020 16:50:07 GMT (14kb)

**[v2]**Wed, 4 Mar 2020 12:44:25 GMT (14kb)

**[v3]**Mon, 22 Feb 2021 15:11:43 GMT (16kb)

Link back to: arXiv, form interface, contact.