Acta Mathematica Hungarica
Volume/Issue:
Volume 126: Issue 1-2
Non-L1 functions with rotation sets of Hausdorff dimension one
Author:
Z. Buczolich Eötvös Loránd University Department of Analysis Pázmány Péter Sétány 1/c 1117 Budapest Hungary

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Pages:
23–50
Online Publication Date:
11 Nov 2009
Publication Date:
01 Jan 2010
Article Category:
Research Article
DOI:
https://doi.org/10.1007/s10474-009-8204-0
Keywords:
rotation; ergodic average; continued fraction; Hausdor-dimension; primary 37A30; secondary 28A78, 28D05, 37E10
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Abstract  

Suppose that f: ℝ → ℝ is a given measurable function, periodic by 1. For an α ∈ ℝ put Mnαf(x) = 1/n+1 Σk=0nf(x + kα). Let Γf denote the set of those α’s in (0;1) for which Mnαf(x) converges for almost every x ∈ ℝ. We call Γf the rotation set of f. We proved earlier that from |Γf| > 0 it follows that f is integrable on [0; 1], and hence, by Birkhoff’s Ergodic Theorem all α ∈ [0; 1] belongs to Γf. However, Γf\ℚ can be dense (even c-dense) for non-L1 functions as well. In this paper we show that there are non-L1 functions for which Γf is of Hausdorff dimension one.

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Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation		 1950
Volumes
per Year		 3
Issues
per Year		 6
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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