### Abstract

We study the statistical properties of piecewise smooth maps on a circle, with a finite number of critical and singular points with an unbounded derivative, such that the derivative goes like the inverse of the distance to the singular points. We write down a simple set of conditions, and show that when these conditions are met, there exist an absolutely continuous invariant probability measure with exponential decay of correlations. We also rule out the existence of nontrivial coboundary, and obtain a positive variance in the central limit theorem for any nonconstant Hölder continuous observable. Our results apply to a positive measure set of nonuniformly expanding maps on the circle considered by Takahasi and Wang (2012 Nonlinearity 25 533).

Original language | English |
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Pages (from-to) | 551-567 |

Number of pages | 17 |

Journal | Nonlinearity |

Volume | 25 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Feb |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Applied Mathematics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**Statistical properties of nonuniformly expanding 1D maps with logarithmic singularities.** / Takahasi, Hiroki.

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 25, no. 2, pp. 551-567. https://doi.org/10.1088/0951-7715/25/2/551

}

TY - JOUR

T1 - Statistical properties of nonuniformly expanding 1D maps with logarithmic singularities

AU - Takahasi, Hiroki

PY - 2012/2

Y1 - 2012/2

N2 - We study the statistical properties of piecewise smooth maps on a circle, with a finite number of critical and singular points with an unbounded derivative, such that the derivative goes like the inverse of the distance to the singular points. We write down a simple set of conditions, and show that when these conditions are met, there exist an absolutely continuous invariant probability measure with exponential decay of correlations. We also rule out the existence of nontrivial coboundary, and obtain a positive variance in the central limit theorem for any nonconstant Hölder continuous observable. Our results apply to a positive measure set of nonuniformly expanding maps on the circle considered by Takahasi and Wang (2012 Nonlinearity 25 533).

AB - We study the statistical properties of piecewise smooth maps on a circle, with a finite number of critical and singular points with an unbounded derivative, such that the derivative goes like the inverse of the distance to the singular points. We write down a simple set of conditions, and show that when these conditions are met, there exist an absolutely continuous invariant probability measure with exponential decay of correlations. We also rule out the existence of nontrivial coboundary, and obtain a positive variance in the central limit theorem for any nonconstant Hölder continuous observable. Our results apply to a positive measure set of nonuniformly expanding maps on the circle considered by Takahasi and Wang (2012 Nonlinearity 25 533).

UR - http://www.scopus.com/inward/record.url?scp=84856157344&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856157344&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/25/2/551

DO - 10.1088/0951-7715/25/2/551

M3 - Article

AN - SCOPUS:84856157344

VL - 25

SP - 551

EP - 567

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 2

ER -