Geometric Growth for Anosov Maps on the 3 Torus

Abstract

We prove that for Anosov maps of the 3-torus if the Lyapunov exponents of absolutely continuous measures in every direction are equal to the geometric growth of the invariant foliations then f is C 1 conjugated to its linear part.Then h * m = λ E A so applying Proposition 2.3, the measure h * m is absolutely continuous in the E A direction with ∆ A (y, t) = 1 (because the Jacobian is constant for the linear map).We need to prove that, restricted to W, h preserves the density ρ, and for every t ∈ W(y),is C r−1 with r > 1, for r = 1 we need f to be at least C 1+α , for some α > 0.The proofs of the next lemmas can be found in [3], for completeness we redo the proofs here.

A note on rigidity of Anosov diffeomorphisms of the three torus

“…Our result is similar in spirit to theirs, although it is announced under different hypothesis and implies their result as a corollary (it does not mean necessarily being stronger). We also refer to M. Poletti [12] result which is in the same spirit.…”

“…See [12], remark 2.5. For our purposes we need to relate conjugacy and absolute continuity with the Lyapunov exponents of the diffeomorphisms.…”

Lyapunov exponents and rigidity of Anosov automorphisms and skew products

“…The assumption in [59] is that f is a C r (r ≥ 2) volume preserving Anosov diffeomorphism isotopic to L, also partially hyperbolic, the center (or weak unstable) foliation is absolutely continuous and the center Lyapunov exponent is the same as the one for L, and the conclusion is that the conjugacy is C 1 . Also a partial result in this direction was obtained by Poletti in [65].…”

The Fourth Fundamental Form of the Torus Hypersurface

“…Surfaces and hypersurfaces have been worked by the mathematicians for centuries. We see some new papers about torus surfaces and torus hypersurfaces in the literature such as [2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”