Continuity of SRB measure and entropy for Benedicks-Carleson quadratic maps

• Type: Article in International Scientific Journal
• Date: 2005

Authors

freitas, jm (Author) FCUP

Journal

Title: Nonlinearity

Vol. 18

Pages: 831-854

ISSN: 0951-7715

Publisher: Institute of Physics Publishing

Indexing

ISI Web of Knowledge - 21 Citations

ISI Web of Science

Scopus - 15 Citations

Scientific classification

FOS: Natural sciences > Mathematics

CORDIS: Physical sciences > Mathematics > Chaos theory

Other information

Authenticus ID: P-000-4KQ

DOI: 10.1088/0951-7715/18/2/019

Abstract (EN): We consider the quadratic family of maps given by f(a)(x) = 1 - ax(2) on I = [-1, 1], for the Benedicks-Carleson parameters. On this positive Lebesgue measure set of parameters close to a = 2, f(a) presents an exponential growth of the derivative along the orbit of the critical point and has an absolutely continuous Sinai-Ruelle-Bowen (SRB) invariant measure. We show that the volume of the set of points of I which, at a given time, fail to present an exponential growth of the derivative, decays exponentially as time passes. We also show that the set of points of I that are not slowly recurrent to the critical set decays sub-exponentially. As a consequence, we obtain continuous variation of the SRB measures and associated metric entropies with the parameter on the referred set. For this purpose, we elaborate on the Benedicks-Carleson techniques in the phase space setting.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 24