Statistical properties of nonuniformly expanding 1D maps with logarithmic singularities

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 languageEnglish
Pages (from-to)551-567
Number of pages17
JournalNonlinearity
Volume25
Issue number2
DOIs
Publication statusPublished - 2012 Feb
Externally publishedYes

Fingerprint

Expanding Maps
Singular Point
Statistical property
Logarithmic
Circle
Singularity
Derivatives
Derivative
Decay of Correlations
Absolutely Continuous
Exponential Decay
Invariant Measure
Central limit theorem
Probability Measure
Critical point
critical point
theorems
nonlinearity
Nonlinearity
decay

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.

In: Nonlinearity, Vol. 25, No. 2, 02.2012, p. 551-567.

Research output: Contribution to journalArticle

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