Abstract
We define new distortion quantities for diffeomorphisms of theEuclidean plane andstudy their properties. In particular, we obtain composition rulesfor these quantities analogous to standard rules for maps of aninterval. Our results apply to maps with unbounded derivatives andhave important applications in the theory of SRB measures for surfacediffeomorphisms.
Highlights
We define new distortion quantities for diffeomorphisms of the Euclidean plane and study their properties
One often is interested in the existence and properties of so-called natural or SRB measures
A fundamental tool in the study of all of these concepts is the investigation of quantitative objects associated to the orbits of submanifolds γ which are expanded from time to time under iteration by the map f
Summary
Formula (3) can be used to show (e.g. as in [6] and [4]) that if f1, f2, . . . is a sequence of expanding diffeomorphisms such that Θ(fi) ≤ K for all i and the lengths of the images of fi are bounded below, there is a constant K1 such that for every n ≥ 1, we have. We say that g Markov precedes f , written f ≻ g or g ≺ f , if Eg′ Ef is a full-width subrectangle of Ef and g−1(Eg′ Ef ) is a full-height subrectangle of Eg. In that case the map h = f ◦ g is a C2 diffeomorphism from the rectangle Eh = g−1(Eg′ Ef ) onto its image Eh′. Let E be a subset of R2, let f : E → E′ be a diffeomorphism, and let C be a separated cone field pair on E E′, with C = (Cu, Cs). It will be convenient to consider analogs of the distortion like quantity Ψ2 above using affine coordinates in which certain maps have diagonal Jacobian matrices.
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