Erwan Lanneau, Stefano Marmi, Alexandra Skripchenko
In the current note we extend results by Marmi, Moussa and Yoccoz about cohomological equations for interval exchange transformations to irreducible linear involutions.
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Let $ϕ:\R^2\to\R^2$ be an orientation--preserving $C^1$ involution such that $ϕ(0)=0$ and let ${\rm Spc}\,(ϕ)=\{{\rm Eigenvalues\,\,of}\,\, Dϕ(p)\mid p\in\R^2\}$. We prove that if ${\rm Spc}\,{(ϕ)}\subset\R$ or ${\rm Spc}\,(ϕ)\cap [1,1+ε)=\emptyset$ for some $ε>0$ then $ϕ$ is globally $C^1$ conjugate to the linear involution $Dϕ(0)$ via the conjugacy $h=(I+Dϕ(0)ϕ)/2$, where $I:\R^2\to\R^2$ is the identity map. Similarly, if $ϕ$ is an orientation-reversing $C^1$ involution such that $ϕ(0)=0$ and ${\rm Trace}\,\big(Dϕ(0)Dϕ(p)\big)>-1 $ for all $p\in\R^2$ then $ϕ$ is globally $C^1$ conjugate to t…
Weak Mixing Property for Linear Involutions
本文研究了线性反交换的弱混合性质，利用Kontsevich-Zorich cocycle的最新进展，证明了动态不可约广义排列对应的线性反交换在几乎所有可接受参数下具有弱混合性。
We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of $\mathbb{R}^{d}$. We use this to characterize closed negatively curved Rie…
Topological weak mixing for certain linear involutions of hyperelliptic type
研究超椭圆型某些线性对合的拓扑弱混合问题，通过考虑特定广义置换\(\sigma_{s,r}\)，证明了由其劳齐类中广义置换定义的典型线性对合的自然符号编码在有至少一个简单字母且相关亏格足够大时是拓扑弱混合的。
We prove that the solutions of the cohomological equation for Roth type interval exchange maps are Hölder continuous provided that the datum is of class $C^r$ with $r>1$ and belongs to a finite-codimension linear subspace.
We provide a general argument for the failure of Anosov-Katok-like constructions (as in \cite{AFKo2015} and \cite{NKInvDist}) to produce Cohomologically Rigid diffeomorphisms in manifolds other than tori. A $C^{\infty }$ smooth diffeomorphism $f $ of a compact manifold $M$ is Cohomologically Rigid iff the equation, known as Linear Cohomological one, \begin{equation*} ψ\circ f - ψ= φ\end{equation*} admits a $C^{\infty }$ smooth solution $ψ$ for every $φ$ in a codimension $1$ closed subspace of $C^{\infty } (M, \mathbb{C} )$. As an application, we show that no Cohomologically Rigid diffeomorphis…
We investigate the natural codings of linear involutions. We deduce from the geometric representation of linear involutions as Poincaré maps of measured foliations a suitable definition of return words which yields that the set of first return words to a given word is a symmetric basis of the free group on the underlying alphabet $A$. The set of first return words with respect to a subgroup of finite index $G$ of the free group on $A$ is also proved to be a symmetric basis of $G$.
An element of a group is \emph{reversible} if it is conjugate to its own inverse, and it is \emph{strongly reversible} if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be expressed as a composite of two involutions. In this paper the reversible maps, the strongly reversible maps, and those maps that can be expressed as a composite of involutions are determined in certain groups of piecewise linear homeomorphisms of the real line.
Cohomological Equation for Robotic Screw Motion on the Lie Group SE(3)
本文研究了SE(3)上螺旋运动的上同调方程，结合傅里叶分析与彼得-魏尔理论，揭示了刚体运动中障碍现象的数学描述。
We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group.
We study the discrete cohomological equation of a hyperbolic affine automorphism Y of the torus (whose linear part is not necessarily diagonalisable). More precisely ; if d is the cobord operator defined by : d (h) = h-hoY for every element h of the Fréchet space E of the differentiable functions on the torus, we show that the image Im(d) is a closed of E and that consequently the space of cohomology H^1(Y; E):=E/Im(d) is a nontrivial Fréchet space. We also prove the existence of a continuous linear operator L defined from Im(d) to E such that for every element g of Im(d), the image f = L(g) i…
A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch provide a subclass of linear involutions. We call such linear involutions non-classical interval exchanges. They are related to measured foliations on orientable flat surfaces. Non-classical interval exchanges can be studied as a dynamical system by considering Rauzy induction in this context. This gives a refinement process on the parameter space similar to Kerckhoff's simplicial systems. We show that the refinement process gives an …
We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter, and we investigate the question of their quasianalyticity. This equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue $q$ of the linear part. This problem has been first investigated by Arnol'd and Herman. Herman raised the question whether the solutions of the cohomological equation had a quasianalytic dependence on the parameter. Indeed they…
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