Automatic sequences as good weights for ergodic theorems

Tanja Eisner; Jakub Konieczny

doi:10.3934/dcds.2018178

Article Contents

2018, Volume 38, Issue 8: 4087-4115. Doi: 10.3934/dcds.2018178

This issue Previous Article The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4 Next Article Lower spectral radius and spectral mapping theorem for suprema preserving mappings

Automatic sequences as good weights for ergodic theorems

Tanja Eisner^1, and
Jakub Konieczny^2,3,

1.
Institute of Mathematics, University of Leipzig, P.O. Box 100 920, 04009 Leipzig, Germany
2.
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel
3.
Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Łojasiewicza 6, 30-348 Kraków, Poland

Received: October 31, 2017

Revised: February 28, 2018

Published: May 2018

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Abstract

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems. We show that automatic sequences are good weights in $ L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $ L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $ L^r$, $ r>1$.

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Mathematics Subject Classification: Primary: 11B85, 37A30; Secondary: 47A35, 68Q45.

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