Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

Stefano Luzzatto, Hiroki Takahasi

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We formulate and prove a Jakobson-Benedicks-Carleson-type theorem on the occurrence of non-uniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we show that the set of parameters corresponding to maps in the quadratic family fa(x) ≤ x2 - a which have an absolutely continuous invariant probability measure is at least 10-5000.

Original languageEnglish
Article number013
Pages (from-to)1657-1695
Number of pages39
JournalNonlinearity
Volume19
Issue number7
DOIs
Publication statusPublished - 2006 Jul 1
Externally publishedYes

Fingerprint

Nonuniform Hyperbolicity
One-dimensional Maps
occurrences
Stochastic Dynamics
Absolutely Continuous
Invariant Measure
Probability Measure
theorems
Lower bound
Theorem
Family

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps. / Luzzatto, Stefano; Takahasi, Hiroki.

In: Nonlinearity, Vol. 19, No. 7, 013, 01.07.2006, p. 1657-1695.

Research output: Contribution to journalArticle

@article{16fc37df5fb94ae1b8697d20e47a638a,
title = "Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps",
abstract = "We formulate and prove a Jakobson-Benedicks-Carleson-type theorem on the occurrence of non-uniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we show that the set of parameters corresponding to maps in the quadratic family fa(x) ≤ x2 - a which have an absolutely continuous invariant probability measure is at least 10-5000.",
author = "Stefano Luzzatto and Hiroki Takahasi",
year = "2006",
month = "7",
day = "1",
doi = "10.1088/0951-7715/19/7/013",
language = "English",
volume = "19",
pages = "1657--1695",
journal = "Nonlinearity",
issn = "0951-7715",
publisher = "IOP Publishing Ltd.",
number = "7",

}

TY - JOUR

T1 - Computable conditions for the occurrence of non-uniform hyperbolicity in families of one-dimensional maps

AU - Luzzatto, Stefano

AU - Takahasi, Hiroki

PY - 2006/7/1

Y1 - 2006/7/1

N2 - We formulate and prove a Jakobson-Benedicks-Carleson-type theorem on the occurrence of non-uniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we show that the set of parameters corresponding to maps in the quadratic family fa(x) ≤ x2 - a which have an absolutely continuous invariant probability measure is at least 10-5000.

AB - We formulate and prove a Jakobson-Benedicks-Carleson-type theorem on the occurrence of non-uniform hyperbolicity (stochastic dynamics) in families of one-dimensional maps, based on computable starting conditions and providing explicit, computable, lower bounds for the measure of the set of selected parameters. As a first application of our results we show that the set of parameters corresponding to maps in the quadratic family fa(x) ≤ x2 - a which have an absolutely continuous invariant probability measure is at least 10-5000.

UR - http://www.scopus.com/inward/record.url?scp=33745388070&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745388070&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/19/7/013

DO - 10.1088/0951-7715/19/7/013

M3 - Article

VL - 19

SP - 1657

EP - 1695

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 7

M1 - 013

ER -