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Communications in Mathematical Physics

, Volume 325, Issue 1, pp 185–257 | Cite as

On the Hyperbolicity of Lorenz Renormalization

  • Marco Martens
  • Björn Winckler
Article

Abstract

We consider infinitely renormalizable Lorenz maps with real critical exponent α > 1 of certain monotone combinatorial types. We prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated two-dimensional strong unstable manifold. For monotone families of Lorenz maps we prove that each infinitely renormalizable combinatorial type has a unique representative within the family. We also prove that each infinitely renormalizable map has no wandering intervals, is ergodic, and has a uniquely ergodic minimal Cantor attractor of measure zero.

Keywords

Composition Operator Periodic Point Unstable Manifold Combinatorial Type Return Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of Symbols

\({\mathcal{K}}\)

The ‘invariant’ set, Sect. 3

\({\bar{\mathcal{K}}}\)

The ‘invariant’ set with pure decompositions, Sect. 8

δ

Upper bound on distortion of diffeomorphic parts for \({f \in \mathcal{K}}\), Sect. 3

\({\varepsilon}\)

Width of right branch for \({f \in \mathcal{K}}\), \({\varepsilon = 1 - c}\), Sect. 3

Ω

Set of monotone types of renormalization, Sect. 3

b0

Lower bound on transfer time of \({c^+_1}\) to C for types in Ω, Sect. 3

D

Derivative operator, Sect. 2

N

Nonlinearity operator, App A

O

Composition operator, Sect. 7

\({\mathcal{R}}\)

Renormalization operator, Sect. 2

M

Derivative of renormalization operator, Sect. 8

S

Schwarzian derivative, App B

Z

Zoom operator, Sect. 2

Q(x)

Standard Lorenz map, Sect. 1

α

Critical exponent, Sect. 1

c

Critical point of a Lorenz map, Sect. 2

u, v

Lorenz map parameters, u = Q(c ), v = 1 − Q(c +), Sect. 2

ϕ, ψ

Diffeomorphic parts of a Lorenz maps, Sect. 2

\({c^-_1,c^+_1}\)

Critical values of a Lorenz map f, \({c^-_1 = f(c^-)}\) , \({c^+_1 = f(c^+)}\) , Sect. 2

a, b

Transfer times of \({c^-_1}\) and \({c^+_1}\) to C Sect. 2

C, L, R

Return interval and its left and right components, C = L ∪ {c} ∪ R, Sect. 2

c

Critical point of renormalization, Sect. 2

u′, v

Parameters of renormalization, Sect. 2

ϕ ′,ψ ′

Diffeomorphic parts of renormalization, Sect. 2

\({c^-_1(\mathcal{R}f)}\)

Critical value of the renormalization, also \({c^+_1(\mathcal{R} f)}\) , Sect. 2

\({\mathcal{D}^k}\)

Set of \({\mathcal{C}^k}\) -diffeomorphisms, Sect. 2

\({\bar{\mathcal{D}}}\)

Set of decompositions, Sect. 7

\({\mathcal{L}^k}\)

Set of \({\mathcal{C}^k}\) -Lorenz maps, Sect. 2

\({\mathcal{L}^S}\)

Set of Lorenz maps with negative Schwarzian derivative, Sect. 2

\({\mathcal{L}_\omega}\)

Set of ω-renormalizable Lorenz maps, Sect. 2

\({\mathcal{L}_{\bar{\omega}}}\)

Set of f such that \({\mathcal{R}^i f}\) is ω i -renormalizable, \({\bar{\omega} = (\omega_0, \omega_1, . . .)}\) , Sect. 2

\({\mathcal{L}_\Omega}\)

Set of ω-renormalizable maps, for some \({\omega \in \Omega}\) , Sect. 2

\({\mathcal{Q}}\)

Set of pure maps, Sect. 7

\({\bar{\mathcal{Q}}}\)

Set of pure decompositions, Sect. 7

\({\asymp}\)

\({g(x) \asymp y}\) iff K −1g(x)/yK, uniformly in x and g, Sect. 9

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA
  2. 2.Department of MathematicsKTHStockholmSweden

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