Edson de Faria, Welington de Melo
We prove that any two real-analytic critical circle maps with cubic critical point and the same irrational rotation number of bounded type are $C^{1+α}$ conjugate for some $α>0$.
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In this paper we establish $C^2$ a-priori bounds for the scaling ratios of critical circle mappings in a form that gives also a compactness property for the renormalization operator.
We survey rigidity results for groups acting on the circle in various settings, from local to global and $C^0$ to smooth. Our primary focus is on actions of surface groups, with the aim of introducing the reader to recent developments and new tools to study groups acting by homeomorphisms.
Let $f$ and $g$ be two circle endomorphisms of degree $d\geq 2$ such that each has bounded geometry, preserves the Lebesgue measure, and fixes $1$. Let $h$ fixing $1$ be the topological conjugacy from $f$ to $g$. That is, $h\circ f=g\circ h$. We prove that $h$ is a symmetric circle homeomorphism if and only if $h=Id$. Many other rigidity results in circle dynamics follow from this very general symmetric rigidity result.
We prove that any two $C^4$ critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a $C^1$ circle diffeomorphism. The conjugacy is $C^{1+α}$ for Lebesgue almost every rotation number.
We prove that any two $C^3$ critical circle maps with the same irrational rotation number of bounded type and the same odd criticality are conjugate to each other by a $C^{1+α}$ circle diffeomorphism, for some universal $α>0$.
We demonstrate the existence of a global attractor A with a Cantor set structure for the renormalization of critical circle mappings. The set A is invariant under a generalized renormalization transformation, whose action on A is conjugate to the two-sided shift.
We prove that if two analytic multicritical circle maps with the same bounded type rotation number are topologically conjugate by a conjugacy which matches the critical points of the two maps while preserving the orders of their criticalities, then the conjugacy necessarily has $C^{1+α}$ regularity, where $α$ depends only on the bound on the type of the rotation number. We then extend this rigidity result to $C^3$-smooth bi-cubic circle maps.
We construct a renormalization operator which acts on analytic circle maps whose critical exponent $α$ is not necessarily an odd integer $2n+1$, $n\in\mathbb N$. When $α=2n+1$, our definition generalizes cylinder renormalization of analytic critical circle maps. In the case when $α$ is close to an odd integer, we prove hyperbolicity of renormalization for maps of bounded type. We use it to prove universality and $C^{1+α}$-rigidity for such maps.
$C^1$ Robust Rigidity for Bi-critical Circle Maps
该研究针对双临界圆周映射的刚性问题，证明满足符号相同、重正化在$C^2$拓扑下指数共同收敛的拓扑共轭双临界圆周映射具有$C^1$共轭性，推进了一维动力系统刚性理论研究。
It is shown that if $f$ and $g$ are any two analytic critical circle mappings with the same irrational rotation number, then the conjugacy that maps the critical point of $f$ to that of $g$ has regularity $C^{1+α}$ at the critical point, with a universal value of $α>0$. As a consequence, a new proof of the hyperbolicity of the full renormalization horseshoe of critical circle maps is given.
We show that a smooth 1-parameter family of foliations by circles of a closed 3-manifold, deforming the foliation whose leaves are the fibers of a circle bundle, is trivial, i.e. all the foliations of the family arise from circle bundles isomorphic to the unperturbed one, if a continuity property of the Seifert leaves of the perturbation holds true. This rigidity property is true for any real analytic 1-parameter family of foliations by circles when the base space of the circle bundle defining the unperturbed foliation is not a torus. The dimensionality hypothesis is discussed in relation to a…
We prove that two $C^r$ critical circle maps with the same rotation number of bounded type are $C^{1+α}$ conjugate for some $α>0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The number $α$ depends only on the rate of convergence. We also give examples of $C^\infty$ critical circle maps with the same rotation number that are not $C^{1+β}$ conjugate for any $β>0$.
We find conditions for two piecewise C^{2+ν} homeomorphisms f and g of the circle to be C^1 conjugate. Besides the restrictions on the combinatorics of the maps (we assume that the maps have bounded combinatorics), and necessary conditions on the one-side derivatives of points where f and g are not differentiable, we also assume zero mean nonlinearity for f and g.
A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes.
We consider order preserving $C^3$ circle maps with a flat piece, Fibonacci rotation number, critical exponents $(\ell_1, \ell_2)$ and negative shwarzian derivative. This paper treat the geometry characteristic of the non-wondering (cantor (fractal)) set from a map of our class. We prove that, for $(\ell_1, \ell_2)$ in $(1,2)^2$, the geometry of system is degenerate (double exponentially fast). As consequences, the renormalization diverges and the geometric (rigidity) class depends on the three couples $(c_u(f), c'_u(f) )$, $( c_+(f), c'_+(f))$ and $(c_s(f), c'_s(f) )$.\vspace{0.5cm}
Let $f$ and $\tilde{f}$ be two circle diffeomorphisms with a break point, with the same irrational rotation number of bounded type, the same size of the break $c$ and satisfying a certain Zygmund type smoothness condition depending on a parameter $γ>2.$ We prove that under a certain condition imposed on the break size $c$, the diffeomorphisms $f$ and $\tilde{f}$ are $C^{1+ω_γ}$-smoothly conjugate to each other, where $ω_γ(δ)=|\log δ|^{-(γ/2-1)}.$
Dynamics and Rigidity through the Lens of Circles
本文通过研究圆排列中的四个自然问题，探讨了无限体积均匀空间的动力学、几何与刚性的相互作用，揭示了均匀动力学的新前沿。
Rigidity for piece-wise smooth circle maps and certain GIETs
本文证明具有一般旋转数、均值为零非线性的分段C^3广义区间交换变换在边界等价且重整化趋近仿射情形下，其共轭映射为C^1，并推广至旋转型组合数据及断点等价的圆微分同胚情形。
Unmarked spectral rigidity of expanding circle maps
本文证明了扩张圆映射在特定条件下长度谱的刚性，表明其无法通过任意小扰动保持长度谱不变。
Rigidity and geometricity for surface group actions on the circle
本文证明了曲面基本群到圆周保向同胚群的刚性表示必为几何表示，从而给出了第一作者相关定理的逆命题。
In this paper we consider two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. It has been shown earlier that such perturbations admit an attractor which is a non-smooth topolgical circle - a "critical" circle. We study conjugacies of the maps that admit such attractors and show that although the maps exhibit universality - they approach a certain normal form when looked at small scales - two maps in general can not be smoothly comjugate on their critical attractors. This result extends the paradigm of "universality but no rigid…
Let $f, g:S^1\to S^1$ be two $C^3$ critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if $h:S^1\to S^1$ is a topological conjugacy between $f$ and $g$ and $h$ maps the critical points of $f$ to the critical points of $g$, then $h$ is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T.~Clark and S.~van Strien \cite{CS}. However, un…
We show that any sufficiently (finitely) smooth $\mathbb Z_2$-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid, i.e. all deformations among domains in the same class which preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak.
We prove a global topological rigidity theorem for locally $C^2$-non-discrete subgroups of the group of real analytic diffeomorphisms of the circle.
In this work we lead with expanding maps of the circle and Anosov diffeomorphisms on $\mathbb{T}^d, d \geq 2.$ We prove that, for these maps, \textit{constant periodic data} imply \textit{same periodic data of these maps and their linearizations}, so in particular we have smooth conjugacy. For expanding maps of the circle and Anosov diffeomorphism on $\mathbb{T}^d, d= 2, 3,$ we have global rigidity. In higher dimensions, $d \geq 4,$ we can establish a result of local rigidity, in several cases. The main tools of this work are celebrated results of rigidity involving same periodic data with lin…
Big mapping class groups and rigidity of the simple circle
本文研究平面去掉康托集后的映射类群在圆上的作用，证明其作用要么平凡，要么半共轭于简单圆上的唯一极小作用。
The rigidity theory for circle homeomophisms with breaks was studied intensively in the last 20 years. It was proved that under mild conditions of the Diophantine type on the rotation number any two $C^{2+α}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-ν}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from th…
In this paper we generalize renormalization theory for analytic critical circle maps with a cubic critical point to the case of maps with an arbitrary odd critical exponent by proving a quasiconformal rigidity statement for renormalizations of such maps.
We show that equidistribution of irrational orbits on the unit circle implies Furstenberg's conjecture.
Rigidity of J-rotational rational maps and critical quasicircle maps
研究J-旋转有理映射和临界准圆映射的刚性性质，证明其在几何有限条件下的Julia集无线场，并展示对Herman环退化问题的组合刚性结果。
Rudolph-Johnson without dynamics: endomorphic rigidity on the circle
本文证明仅凭熵条件即可使圆周上的测度在亚空隙乘子自同态下趋于 Lebesgue 测度，给出有限熵界并推广 Rudolph-Johnson 定理，同时揭示 Lyons 猜想的成立与失败条件。
We construct an increasing sequence of natural numbers $(m_n)_{n=1}^{+\infty}$ with the property that $(m_n þ[1])_{n\geq 1}$ is dense in $\T$ for any $þ\in \R\setminus \Q$, and a continuous measure on the circle $μ$ such that $\lim_{n\to +\infty}\int_{\T}\|m_nθ\|dμ(θ)=0$. Moreover, for every fixed $k\in \N$, the set $\{n\in \N:\,k\nmid m_n \}$ is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.
Harmonic measures and rigidity for surface group actions on the circle
该研究探讨无挠格$\text{PSU}(1,1)$在$S^1$上作用的刚性，通过悬垂丛叶状调和测度方法，得到曲率估计与Gauss-Bonnet公式，精确描述最大欧拉数悬垂叶状结构的调和测度并另证相关刚性定理。
In this article we investigate rigidity properties of integrable area-preserving twist maps of the cylinder. More specifically, we prove that if a deformation of the standard integrable map preserves rotational invariant circles (i.e., homotopically non-trivial invariant curves of the cylinder on which the dynamics is conjugate to a rotation) for any rational rotation number in an open interval (without any further assumption on its size), then the deformation must be trivial. Analogue phenomena for integrable Tonelli Hamiltonian flows in higher dimensions are discussed.
Inspired by the recent results toward Birkhoff conjecture (a rigidity property of billiards in ellipses), we discuss two rigidity properties of conics. The first one concerns symmetries of an analog of polar duality associated with an oval, and the second concerns properties of the circle map associated with an oval and two pencils of lines.
In this work we treat a famous topic in Ergodic Theory and Dynamical Systems: uniformly expanding maps. We relate regularity of expanding maps and conjugacies with Lyapunov exponents, metric and topological entropies for expanding maps of the circle. In Theorem C, we present a result of rigidity of expanding maps on the torus $\mathbb{T}^2,$ after we naturally generalize it in higher dimensions. Here, we present techniques involving ordinary differential equations to study rigidity problems of expanding maps. Along of this work we provide another proof of a Shub-Sullivan's Theorem, concerning …
In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary assumptions are satisfied). The aim of this paper is to conclude this programme in a natural class of $C^3$ mappings. Examples of such rigidity were established previously, but not, for example, for real polynomials with non-real critical points. Our results are also new for analytic mappings. The main new ingredients of the proof in the real analytic case are (i…
A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper we establish this principle for a large class of bicritical circle maps, which are $C^3$ circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo for the case o…
The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in $\mathbb{C}^k$, $k \ge 3.$ First, we study the relation between two polynomial shift-like maps in $\mathbb{C}^k$, $k \ge 3$, that share the same backward and forward Julia sets (or non-escaping sets). Secondly, we study the relationship between any pair of skew products of Hénon maps in $\mathbb{C}^3$ having the same forward and backward Julia sets.
A numerical study of rigidity of hyperbolic splittings in simple two-dimensional maps
本文通过数值模拟研究简单二维解析映射中的双曲分裂刚性，证明其双曲结构非光滑，为光滑动力系统中产生非光滑相空间结构的全局机制提供了例证。
Lyapunov spectrum rigidity and simultaneous linearization for random Anosov diffeomorphisms
本文研究了随机阿诺索夫微分同胚的李雅普诺夫谱刚性，证明了在特定条件下系统可同时线性化，并探讨了正矩阵随机游走的正李雅普诺夫指数刚性。
We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points…
Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.
A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered Anosov flow. The proof uses the universal circle to R-covered foliations.
Rigidity of Kleinian groups via self-joinings
本文通过引入自连接高阶动力学的新视角，证明了在极限集上具有共形边界映射的非Fuchsian Kleinian群的新刚性定理，并回答了McMullen关于零体积四面体顶点映射刚性的问题。
We show that if $G_1$ and $G_2$ are non-solvable groups, then no $C^{1,τ}$ action of $(G_1\times G_2)*\mathbb{Z}$ on $S^1$ is faithful for $τ>0$. As a corollary, if $S$ is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group $\mathrm{Mod}(S)$ with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.
Circle packings, renormalizations and subdivision rules
本文利用 Teichmüller 空间上剥皮映射的迭代，为满足特定细分规则的圆填充发展了重整化理论，证明重整化算子一致收缩，从而完整解决了此类圆填充的存在性与模问题，并证明了其几何不灵活性与组合刚性。
We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume (whenever the manifold is hyperbolic), the dilatation of the monodromy, the injectivity radius, and the bottom eigenvalue of the Laplacian on these mapping tori. We also study mapping tori of punctured surface bundles, and various invariants of their Dehn fillings. We also study …
Local rigidity of actions of isometries on compact real analytic Riemannian manifolds
本文研究解析黎曼流形上等距变换的小解析扰动，在类丢番图条件下证明有限展示群的小扰动解析共轭于原等距群，推广了 Arnold-Herman 定理。
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