There are no sigma-finite absolutely continuous invariant measures for multicritical circle maps

Author(s)

de Faria, Edson ^{[1, 2]} ; Guarino, Pablo ^{[3, 4]}

Affiliation

^[1] Univ Sao Paulo, Inst Matemat & Estat, Sao Paulo - Brazil

^[2] Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

^[3] Univ Fed Fluminense, Inst Matemat & Estat, Niteroi, RJ - Brazil

^[4] Rua Prof Marcos Waldemar de Freitas Reis S-N, Niteroi, RJ - Brazil

• Document type: Journal article
• Source: Nonlinearity; v. 34, n. 10, p. 6727-6749, OCT 2021.
• Web of Science Citations: 0

Abstract

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. (AU)

• FAPESP's process: 16/25053-8 - Dynamics and geometry in low dimensions
• Grantee: André Salles de Carvalho
• Support Opportunities: Research Projects - Thematic Grants