Related papers: Bootstrap for local rigidity of Anosov automorphis…
We consider a hyperbolic toral automorphism $L$ and its $C^1$-small perturbation $f$. It is well-known that $f$ is Anosov and topologically conjugate to $L$, but a conjugacy $H$ is only H\"older continuous in general. We discuss conditions…
We consider an irreducible Anosov automorphism L of a torus T^d such that no three eigenvalues have the same modulus. We show that L is locally rigid, that is, L is C^1 conjugate to any C^1-small perturbation f with the same periodic data.…
We study the regularity of the conjugacy between an Anosov automorphism $L$ of a torus and its small perturbation. We assume that $L$ has no more than two eigenvalues of the same modulus and that $L^4$ is irreducible over $\mathbb Q$. We…
Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\ge3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\mathcal U$ of $L$. The main result establishes $C^{1+\nu}$ regularity of the conjugacy between two Anosov…
Given a $C^0$ conjugacy between two Anosov diffeomorphisms, the matching periodic data problem asks whether this conjugacy is smooth provided spectral data of the diffeomorphisms match at periodic points. We show that if the two $C^0$…
We study regularity of a conjugacy between a hyperbolic or partially hyperbolic toral automorphism $L$ and a $C^\infty$ diffeomorphism $f$ of the torus. For a very weakly irreducible hyperbolic automorphism $L$ we show that any $C^1$…
We study the regularity of a conjugacy $H$ between a hyperbolic toral automorphism $A$ and its smooth perturbation $f$ We show that if $H$ is weakly differentiable then it is $C^{1+H\"older}$ and, if $A$ is also weakly irreducible, then $H$…
In this paper we introduce a new methodology for smooth rigidity of Anosov diffeomorphisms based on "matching functions." The main observation is that under certain bunching assumptions on the diffeomorphism the periodic cycle functionals…
In this work we completely classify $C^\infty$ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of…
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…
In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with…
We study local rigidity properties of holomorphic embeddings of real hypersurfaces in $\mathbb C^2$ into real hypersurfaces in $\mathbb C^3$ and show that infinitesimal conditions imply actual local rigidity in a number of (important)…
We obtain smooth conjugacy between non-necessarily special Anosov endomorphisms in the conservative case. Among other results, we prove that a strongly special $C^{\infty}-$Anosov endomorphism of $\mathbb{T}^2$ and its linearization are…
We study the regularity of a conjugacy between an Anosov automorphism $L$ of a nilmanifold $N/\Gamma$ and a volume-preserving, $C^1$-small perturbation $f$. We say that $L$ is locally Lyapunov spectrum rigid if this conjugacy is $C^{1+}$…
De la Llave's examples are Anosov diffeomorphisms on the four-torus $\mathbb{T}^4$ with constant Lyapunov spectrum, yet they are not $C^{1}$-conjugate to the linear model or to each other. Nevertheless, we show that such examples are…
We consider a hyperbolic automorphism $A\colon\mathbb T^3\to\mathbb T^3$ of the 3-torus whose 2-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold $A$ into two one-parameter families of Anosov…
We show that, given a real or complex hyperbolic metric $g_0$ on a closed manifold $M$ of dimension $n\geq 3$, there exists a neighborhood $\mathcal U$ of $g_0$ in the space of negatively curved metrics such that for any $g\in \mathcal U$,…
We consider two transitive $3$-dimensional Anosov flows which do not preserve volume and which are continuously conjugate to each other. Then, disregarding certain exceptional cases, such as flows with $C^1$ regular stable or unstable…
In the present work we obtain rigidity results analysing the set of regular points, in the sense of Oseledec's Theorem. It is presented a study on the possibility of an Anosov diffeomorphisms having all Lyapunov exponents defined…
We prove that the deformation space AH(M) of marked hyperbolic 3-manifolds homotopy equivalent to a fixed compact 3-manifold M with incompressible boundary is locally connected at minimally parabolic points. Moreover, spaces of Kleinian…