Leonardo Dinamarca, Maximiliano Escayola
We improve a recent construction of Andrés Navas to produce the first examples of $C^2$-undistorted diffeomorphisms of the interval that are $C^{1+α}$-distorted (for every $α< 1$). We do this via explicit computations due to the failure of an extension to class $C^{1+α}$ of a classical lemma related to the work of Nancy Kopell.
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We build the first examples of diffeomorphisms that are distorted in a group of $C^r$ diffeomorphisms yet undistorted in the corresponding group of $C^s$ diffeomorphisms, where $r < s$. This explicit construction is performed for the closed interval, $r = 1$ and $s = 2$.
All, most, some? On diffeomorphisms of the interval that are distorted and/or conjugate to powers of themselves
本研究探讨区间微分同胚共轭于自身幂次的问题，分析了不同正则性下相关集合的稠密性与泛型性等性质，补充了区间微分同胚群扭曲元素的相关结果。
How (not) to prove (un)distortion for diffeomorphisms of one-manifolds
研究给定一维流形\(M\)及\(1\leq r\lt s\leq\infty\)，是否存在\(C^s\)保向紧支微分同胚在\(\mathrm{Diff}_{c,+}^s(M)\)中无扭曲但在更大的\(C^r\)微分同胚群中有扭曲。分析了不同\((r,s)\)情况及相关阻碍。
We extend to the critical (intermediate) regularity several results concerning rigidity for centralizers and group actions on the interval.
Finitely generated groups of diffeomorphisms with prescribed critical regularity in every dimension
该论文构造了嵌入到指定临界正则性 $C^\alpha$ 微分同胚群但不嵌入更高正则性群的有限生成群，适用于球面和实射影空间。
On residues and conjugacies for germs of 1-D parabolic diffeomorphisms in finite regularity
本文研究固定原点的实直线非平坦微分同胚芽的共轭类，基于Takens和Yoccoz工作，证明低正则共轭下留数不变性，并推广Schwarzian导数值在$C^2$抛物共轭下不变而在$C^1$下可变的结果。
According to the classical Plante-Thurston Theorem, all nilpotent groups of $C^2$-diffeomorphisms of the closed interval are Abelian. Using techniques coming from the works of Denjoy and Pixton, Farb and Franks constructed a faithful action by $C^1$-diffeomorphisms of $[0,1]$ for every finitely-generated, torsion-free, non-Abelian nilpotent group. In this work, we give a version of this construction that is sharp in what concerns the Hölder regularity of the derivatives. Half of the proof relies on results on random paths on Heisenberg-like groups that are interesting by themselves.
Relation between irrationality and regularity for $ C^1 $ conjugacy of $ C^2 $ circle diffeomorphisms to rigid rotations
本文研究了C^2圆微分同胚与刚性旋转的C^1共轭性中无理性和正则性的关系，推导了Denjoy型不等式并首次建立了连续性与无理性的显式关联。
Let $M$ be the circle or a compact interval, and let $α=k+τ\ge1$ be a real number such that $k=\lfloor α\rfloor$. We write $\mathrm{Diff}_+^α(M)$ for the group of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives are Hölder continuous with exponent $τ$. If $α\ge1$, we prove that there exists a finitely generated subgroup $G_α\le\mathrm{Diff}_+^α(M)$ with the property that $G_α$ admits no injective homomorphisms into $\mathrm{Diff}_+^β(M)$ for all $β>α$. If $α>1$, we also show the dual result: there exists a finitely generated group $H_α\le\bigcap_{β<α}\mathrm{Diff}_+^β(M)$ with the prope…
In this paper we consider the conjugacy classes of diffeomorphisms of the interval, endowed with the $C^1$-topology. We present several results in the spirit of the one below : Given two diffeomorphisms $f,g$ of the interval $[0;1]$ without hyperbolic fixed point, we give a complete answer to the two following questions: - under what conditions does there exist a sequence of smooth conjugates $h_n f h_n^{-1}$ of $f$ tending to $g$ in the $C^1$-topology ? - under what conditions does there exist a continuous path of $C^1$-diffeomorphisms $h_t$ such that $h_t f h_t^{-1}$ tends to $g$ in the $C^1…
Given any smooth Anosov map we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that in the case of expanding maps it reduces exactly to the usual space of functions of bounded variation which has proven particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations which provides new information that could prove useful in treating hyperbolic systems with singularities.
Negative regularity mixing for random volume preserving diffeomorphisms
本文研究随机保体积微分同胚的负正则性指数混合，给出通用判据并应用于随机微分同胚、SDE 及随机不可压缩 Navier-Stokes 平流扩散，证明理想标量湍流存在唯一平稳测度。
We establish a theory for the existence and regularity of solutions to the cohomological equation over an accessible, partially hyperbolic diffeomorphism. As a by-product of our techniques, we show that for $r>1$, any $C^r$ homogeneous, locally compact submanifold of a $C^r$ manifold is in fact a $C^r$ submanifold.
We consider families of diffeomorphisms with dominated splittings and preserving a Borel probability measure, and we study the regularity of the Lyapunov exponents associated to the invariant bundles with respect to the parameter. We obtain that the regularity is at least the sum of the regularities of the two invariant bundles (for regularities in $[0,1]$), and under suitable conditions we obtain formulas for the derivatives. Similar results are obtained for families of flows, and for the case when the invariant measure depends on the map. We also obtain several applications. Near the time on…
$C^1$ linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions. However, $C^1$ smoothness is not enough to guarantee $C^1$ linearization. For $C^{1,1}$ hyperbolic diffeomorphisms on Banach spaces $C^1$ linearization was proved under a gap condition together with a band condition of the spectrum. In this paper, the result of $C^1$ linearization in Banach spaces is strengthened to $C^{1,β}$ linearization with a constant $β>0$ under a weaker band condition by a decomposition with invariant foliat…
We study the regularity of exceptional actions of groups by $C^{1,α}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $α$. Let $G$ be a finitely generated group admitting a $C^{1,α}$ action $ρ$ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $G$ has spherical growth bounded by $c n^{d-1}$ and if the function $1/α^d$ is integrable near zero, then under som…
We consider group-valued cocycles over a partially hyperbolic diffeomorphism which is accessible volume-preserving and center bunched. We study cocycles with values in the group of invertible continuous linear operators on a Banach space. We describe properties of holonomies for fiber bunched cocycles and establish their Holder regularity. We also study cohomology of cocycles and its connection with holonomies. We obtain a result on regularity of a measurable conjugacy, as well as a necessary and sufficient condition for existence of a contionuous conjugacy between two cocycles.
We determine the largest α for which the nilpotent group of 4-by-4 triangular matrices with integer coefficients and 1 in the diagonal embeds into the group of C1+α diffeomorphism of the closed interval.
In these expository notes, we give a proof of regularity of Anosov splitting for Anosov diffeomorphisms in a torus. We also generalize the idea to higher dimensions and to Anosov flows.
We relate the Mather invariant of diffeomorphisms of the (closed) interval to their asymptotic distortion. For maps with only parabolic fixed points, we show that the former is trivial if and only if the latter vanishes. As a consequence, we obtain that such a diffeomorphism of the interval with no fixed point in the interior contains the identity in the closure of its C^{1+bv} conjugacy class if and only if it is the time-1 map of a C^1 vector field. A corollary of this is that diffeomorphisms that do not arise from vector fields are undistorted in the whole group of interval interval diffeom…
Quantitative Estimates on Invariant Manifolds for Surface Diffeomorphisms
本文通过显式不等式量化二维耗散系统轨道的正则性，并将其与局部动力学的拟线性化联系起来，从而在不依赖不变测度的情况下证明正则轨道具有一致有界几何结构的光滑稳定流形和中心流形，为Hénon-like映射重整化理论奠定技术基础。
Several perturbation tools are established in the volume preserving setting allowing for the pasting, extension, localized smoothing and local linearization of vector fields. The pasting and local linearization hold in all classes of regularity ranging from $C^{1}$ to $C^{\infty}$ (Hölder included). For diffeomorphisms, a conservative linearized version of Franks lemma is proved in the $C^{r,α}$ ($r\in\mathbb{Z}^+$, $0<α<1$) and $C^{\infty}$ settings, the resulting diffeomorphism having the same regularity as the original one.
Complete characterization of planar $C^1$ linearizability and optimal conjugacy regularity
本文完全刻画了平面微分同胚的局部$C^1$线性化条件，给出三个最优Dini型阈值，并确定双曲情形下线性化共轭的最优正则性。
Regularity of conjugacies of linearizable generalized interval exchange transformations
本文研究可线性化广义区间交换变换的共轭正则性，利用加速 Rauzy-Veech 归纳的重整化算子证明在指数快速收敛条件下共轭映射具有 C^{1+α} 正则性，并将亏格二刚性结果从 C^1 提升到 C^{1+α}。
On the sharpness of Denjoy's theorem
本研究构造了一类正则性为C^{1+ω}、具无理旋转数且含游荡区间的圆周微分同胚，证明了Denjoy定理在正则性上的锐度，还解决了ω(t)=tlog(1/t)情形下自Herman 1979年工作遗留的公开问题。
On the critical regularity of nilpotent groups acting on the interval: the metabelian case
本文研究无挠有限生成幂零亚阿贝尔群在区间上保定向微分同胚群中的嵌入正则性，证明临界正则性由商群无挠秩决定，并以海森堡群为例给出精确临界值。
We study spatial discretizations of dynamical systems: is it possible to recover some dynamical features of a system from numerical simulations? Here, we tackle this issue for the simplest algorithm possible: we compute long segments of orbits with a fixed number of digits. We show that for every $r\ge 1$, the dynamics of the discretizations of a $C^r$ generic conservative diffeomorphism of the torus is very different from that observed in the $C^0$ regularity. The proof of our results involves in particular a local-global formula for discretizations, as well as a study of the corresponding li…
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 $C^{1+ν}$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations. We elaborate on the example of de la Llave of two Anosov systems on $\mathbb T^4$ with the same co…
Extremal distributions of partially hyperbolic systems: the Lipschitz threshold
研究部分双曲系统极值分布的正则性临界现象，证明当极值分布为Lipschitz时自动为C^∞，并应用于刚性问题，如研究保守设定下的联合积分性与u-Gibbs测度的刚性结果。
On the regularity of time-delayed embeddings with self-intersections
本文研究时滞嵌入在允许零测集自交的概率情形下的正则性，证明在维数条件下典型观测函数可使时滞映射在满测集上单射且具有逐点 Lipschitz 逆，并应用于 Lyapunov 指数逼近与预测误差估计。
Fourier decay of equilibrium states and the Fibonacci Hamiltonian
研究曲面上非线性保面积公理 A 微分同胚平衡态的傅里叶衰减，利用和积现象及推广 Tsujii - Zhang 方法，得到自共形测度下傅里叶维数为正等结果，还证明了关于圆扩张和斐波那契哈密顿量的推论。
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