Misha Guysinsky | Searcharxiv

arXiv2014

Livšic Theorem for Banach Rings

We prove the Livšic Theorem for Hölder continuous cocycles with values in Banach rings. We consider a transitive homeomorphism ${σ:X\to X}$ that satisfies the Anosov Closing Lemma, and a Hölder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. We show that it is a coboundary with a Hölder continuous transition function if and only if ${a(σ^{n-1}p)\ldots a(σp)a(p)=e}$ for each periodic point $p=σ^n p$.

math.DS↗

arXiv2008

Differentiable Conjugacy of Anosov Diffeomorphisms on Three Dimensional Torus

We consider two C^2 Anosov diffeomorphisms in a C^1 neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum. We prove that they are C^1+ conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues.

math.DS↗