Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps

Viviane Baladi; Michael Benedicks; Daniel Schnellmann

doi:10.1007/s00222-014-0554-8

Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps

Viviane Baladi, Michael Benedicks, Daniel Schnellmann

Institut for Matematiske Fag

23 Citationer (Scopus)

Abstract

We consider C² families t ↦ft of C⁴ nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure μ_t of f_t, as a function of t on the set of Collet–Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters Δ, and, for each t0∈Δ, a set Δ0⊂Δ of polynomially recurrent parameters containing t0 as a Lebesgue density point, and constants C≥1, >4Γ>4, so that, for every 1/2-Hölder function A, (Formula presented.). In addition, for all t∈Δ0, the renormalisation period Pt≤Pt0, and there are uniform bounds on the rates of mixing of (Formula presented.) for all t with P_t=P_t0. If f_t(x)=tx(1-x), the set Δ contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz–Thurston parameter t0, we find a set of CE parameters ΔMT′ accumulating at t0, a constant C≥1, and a C∞ function A0, so that (Formula presented.).

Originalsprog	Engelsk
Tidsskrift	Inventiones Mathematicae
Vol/bind	201
Udgave nummer	3
Sider (fra-til)	773-844
ISSN	0020-9910
DOI	https://doi.org/10.1007/s00222-014-0554-8
Status	Udgivet - 22 sep. 2015

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10.1007/s00222-014-0554-8

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title = "Whitney–H{\"o}lder continuity of the SRB measure for transversal families of smooth unimodal maps",

abstract = "We consider C2 families t ↦ft of C4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure μt of ft, as a function of t on the set of Collet–Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters Δ, and, for each t0∈Δ, a set Δ0⊂Δ of polynomially recurrent parameters containing t0 as a Lebesgue density point, and constants C≥1, >4Γ>4, so that, for every 1/2-H{\"o}lder function A, (Formula presented.). In addition, for all t∈Δ0, the renormalisation period Pt≤Pt0, and there are uniform bounds on the rates of mixing of (Formula presented.) for all t with Pt=Pt0. If ft(x)=tx(1-x), the set Δ contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz–Thurston parameter t0, we find a set of CE parameters ΔMT′ accumulating at t0, a constant C≥1, and a C∞ function A0, so that (Formula presented.).",

author = "Viviane Baladi and Michael Benedicks and Daniel Schnellmann",

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T1 - Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps

AU - Baladi, Viviane

AU - Benedicks, Michael

AU - Schnellmann, Daniel

PY - 2015/9/22

Y1 - 2015/9/22

N2 - We consider C2 families t ↦ft of C4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure μt of ft, as a function of t on the set of Collet–Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters Δ, and, for each t0∈Δ, a set Δ0⊂Δ of polynomially recurrent parameters containing t0 as a Lebesgue density point, and constants C≥1, >4Γ>4, so that, for every 1/2-Hölder function A, (Formula presented.). In addition, for all t∈Δ0, the renormalisation period Pt≤Pt0, and there are uniform bounds on the rates of mixing of (Formula presented.) for all t with Pt=Pt0. If ft(x)=tx(1-x), the set Δ contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz–Thurston parameter t0, we find a set of CE parameters ΔMT′ accumulating at t0, a constant C≥1, and a C∞ function A0, so that (Formula presented.).

AB - We consider C2 families t ↦ft of C4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure μt of ft, as a function of t on the set of Collet–Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters Δ, and, for each t0∈Δ, a set Δ0⊂Δ of polynomially recurrent parameters containing t0 as a Lebesgue density point, and constants C≥1, >4Γ>4, so that, for every 1/2-Hölder function A, (Formula presented.). In addition, for all t∈Δ0, the renormalisation period Pt≤Pt0, and there are uniform bounds on the rates of mixing of (Formula presented.) for all t with Pt=Pt0. If ft(x)=tx(1-x), the set Δ contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz–Thurston parameter t0, we find a set of CE parameters ΔMT′ accumulating at t0, a constant C≥1, and a C∞ function A0, so that (Formula presented.).

U2 - 10.1007/s00222-014-0554-8

DO - 10.1007/s00222-014-0554-8

M3 - Journal article

SN - 0020-9910

VL - 201

SP - 773

EP - 844

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -

Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps

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