Journal
NONLINEARITY
Volume 34, Issue 10, Pages 6727-6749Publisher
IOP PUBLISHING LTD
DOI: 10.1088/1361-6544/ac1a02
Keywords
critical circle maps; sigma-finite measures; Katznelson's criterion
Categories
Funding
- 'Projeto Tematico Dinamica em Baixas Dimensoes' FAPESP [2016/25053-8]
- Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq)
- Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior-Brasil (CAPES) [23038.009189/2013-05]
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This paper demonstrates that multicritical circle maps without periodic orbits cannot leave invariant an infinite, sigma-finite invariant measure which is absolutely continuous with respect to Lebesgue measure.
It is well-known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which is purely singular with respect to Lebesgue measure. Can such a map leave invariant an infinite, sigma-finite invariant measure which is absolutely continuous with respect to Lebesgue measure? In this paper, using an old criterion due to Katznelson, we show that the answer to this question is no.
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