Communications in Mathematical Physics

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

On the Hyperbolicity of Lorenz Renormalization

  • Marco Martens
  • Björn Winckler


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.


Composition Operator Periodic Point Unstable Manifold Combinatorial Type Return Interval 
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List of Symbols


The ‘invariant’ set, Sect. 3


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


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


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


Set of monotone types of renormalization, Sect. 3


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


Derivative operator, Sect. 2


Nonlinearity operator, App A


Composition operator, Sect. 7


Renormalization operator, Sect. 2


Derivative of renormalization operator, Sect. 8


Schwarzian derivative, App B


Zoom operator, Sect. 2


Standard Lorenz map, Sect. 1


Critical exponent, Sect. 1


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


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


Critical point of renormalization, Sect. 2

u′, v

Parameters of renormalization, Sect. 2

ϕ ′,ψ ′

Diffeomorphic parts of renormalization, Sect. 2


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


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


Set of decompositions, Sect. 7


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


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


Set of ω-renormalizable Lorenz maps, Sect. 2


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


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


Set of pure maps, Sect. 7


Set of pure decompositions, Sect. 7


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


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© 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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