Growth sequences for circle diffeomorphisms

A generic dimensional property of the invariant measures for circle diffeomorphisms

Given any Liouville number $α$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $α$.

Dimensional characteristics of invariant measures for circle diffeomorphisms

We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number $τ$ there exists a $C^\infty$ circle diffeomorphism with rotation number $τ$ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value $0\le β\le 1$.

The Denjoy type-of argument for quasiperiodically forced circle diffeomorphisms

We carry the argument used in the proof of the Theorem of Denjoy over to the quasiperiodically forced case. Thus we derive that if a system of quasiperiodically forced circle diffeomorphisms with bounded variation of the derivative has no invariant graphs with a certain kind of topological regularity, then the system is topologically transitive.

Derivatives of rotation number of one parameter families of circle diffeomorphisms

We consider the rotation number $ρ(t)$ of a diffeomorphism $f_t=R_t\circ f$, where $R_t$ is the rotation by $t$ and $f$ is an orientation preserving $C^\infty$ diffeomorphism of the circle $S^1$. We shall show that if $ρ(t)$ is irrational $$\limsup_{t'\to t}(ρ(t')-ρ(t))/(t'-t)\geq 1.$$

On the invariant distributions of C^2 circle diffeomorphisms of irrational rotation number

We show that no C^2 circle diffeomorphism of irrational rotation number has invariant 1-distributions other than (scalar multiples of) the invariant measure. We also show that this is false in the C^1 context by giving both minimal and non-minimal examples.

A Denjoy Theorem for commuting circle diffeomorphisms with mixed Holder derivatives

We prove that if d is an integer number bigger than 1 and f_1,...,f_d are commuting circle diffeomorphisms respectively of class C^(1+τ_k), where τ_1 + ... + τ_k > 1, then these maps are simultaneously conjugate to rotations provided that their rotation numbers are independent over the rationals.

Jean-Christophe Yoccoz and the theory of circle diffeomorphisms

Our aim in this note is to present some of the crucial contributions of Jean-Christophe Yoccoz to the theory of circle diffeomorphisms. We start with a short historical account before exposing Yoccoz' work. Then we give a brief description of the main conceptual and technical tools of the theory, with a focus on describing Yoccoz' work and contributions.

Topological conjugacy of circle diffeomorphisms

The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy. In 1985, one of us (Sullivan) gave a new criterion. There is an example satisfying Denjoy's bounded variation condition rather than Sullivan's Zygmund condition and vice versa. This paper will give the third criterion which is implied by either of the above criteria.

Locally discrete groups of analytic diffeomorphisms of the circle

We show that a finitely generated group of analytic diffeomorphisms that is expanding and locally discrete in the analytic category is analytically conjugate to a uniform lattice of a finite covering of the group of projective maps of the projective line, acting naturally on the corresponding finite covering of the projective line.

Maximal entropy measures of diffeomorphisms of circle fiber bundles

We characterize the maximal entropy measures of partially hyperbolic C^2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons. In the special case of 3-dimensional nilmanifolds other than the torus, this entails the following dichotomy: either the diffeomorphism is a rotation extension of an Anosov diffeomorphism -- in which case there is a unique maximal measure, with full support and zero center Lyapunov exponents -- or there exist exactly two ergodic maximal measures, both hyperbolic and whose center Lyapunov …

Rigidity for circle diffeomorphisms with breaks satisfying a Zygmund smoothness condition

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 $ω_γ(δ)=|\log δ|^{-(γ/2-1)}.$

广义的圆微分同胚族具有许多共存的周期轨道

Generic families of circle diffeomorphisms have many coexisting periodic orbits

本文证明了对于广义的圆微分同胚族，每个对应于无理旋转数的参数值都能被具有任意大有限周期轨道数的参数值近似，表明无理旋转数出现的族不弱结构稳定。同时证明了任何局部残余集的一参数族非常数旋转数会给出弱等价类的连续集合。

Classification of generic semigroup actions of circle diffeomorphisms

We study topological properties of semi-group actions on the circle by orientation-preserving homeomorhisms. We prove that a generic action either possesses a forward-invariant interval-domain (i.e. a finite union of disjoint circle arcs), or is two-sided minimal (and, moreover, by a small perturbation one can create a map with arbitrary rotation number in the semigroup). For the minimal case, we also study conditions required for global synchronization.

Orbit equivalence types of circle diffeomorphisms with a Liouville rotation number

This paper is concerned about the orbit equivalence types of $C^\infty$ diffeomorphisms of $S^1$ seen as nonsingular automorphisms of $(S^1,m)$, where $m$ is the Lebesgue measure. Given any Liouville number $α$, it is shown that each of the subspace formed by type ${\rm II}_1$, ${\rm II}_\infty$, ${\rm III}_λ$ ($λ>1$), ${\rm III}_\infty$ and ${\rm III}_0$ diffeomorphisms are $C^\infty$-dense in the space of the orientation preserving $C^\infty$ diffeomorphisms with rotation number $α$.

A circle diffeomorphism with breaks that is smoothly linearizable

In this paper we answer positively a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case when its breaks are lying on pairwise distinct trajectories. An example constructed is a piecewise linear circle homeomorphism that has four break points lying on distinct trajectories, and whose invariant measure is absolutely continuous w.r.t.\ the Lebesgue measure. The irrational rotation number for our example can be chosen a Roth number, but not of bounded type.

具有平坦区间的某些圆周映射的$C^1$-微分同胚类

$C^1$-Diffeomorphism Class of some Circle Maps with a Flat Interval

研究平坦段边界奇点次数不同的圆周映射类，证明其拓扑共轭若为双Lipschitz同胚则必为$C^1$微分同胚，从而两类等价。

Renormalization of circle diffeomorphisms with a break-type singularity

Let $f$ be an orientation-preserving circle diffeomorphism with irrational rotation number and with a break point $ξ_{0},$ that is, its derivative $f'$ has a jump discontinuity at this point. Suppose that $f'$ satisfies a certain Zygmund condition dependent on a parameter $γ>0.$ We prove that the renormalizations of $f$ are approximated by Möbius transformations in $C^{1}$-norm if $γ\in (0,1]$ and they are approximated in $C^{2}$-norm if $γ\in (1,+\infty).$ It is also shown, that the coefficients of Möbius transformations get asymptotically linearly dependent.

On the ergodic theory of free group actions by real-analytic circle diffeomorphisms

We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to É. Ghys, G. Hector and D. Sullivan.

Commuting circle diffeomorphisms with their derivatives having mixed moduli of continuity

Let $d\geq 2$ be an integer and let $ω_1,\cdots ,ω_d$ be moduli of continuity in a specified class which contains the moduli of Hölder continuity. Let $f_k$, $k\in\{1,\cdots,d\}$, be $C^{1+ω_k}$ orientation preserving diffeomorphisms of the circle and $f_1,\cdots, f_d$ commute with each other. We prove that if the rotation numbers of $f_k$'s are independent over the rationals and $ω_1(t)\cdotsω_d(t)=tω(t)$ with $\lim_{t\rightarrow 0^+}ω(t)=0$, then $f_1,\cdots,f_d$ are simultaneously (topologically) conjugate to rigid rotations.

圆周微分同胚群作用的维数理论 I：平稳测度

Dimension theory of group actions by circle diffeomorphisms I: Stationary measures

该论文研究圆周光滑随机游走平稳测度的维数性质，为同主题第二部分提供技术工具，还得到相关逼近定理与圆周作用时间反转对称性结论。

The Poisson boundary of a locally discrete group of diffeomorphisms of the circle

We compute the Poisson boundary of locally discrete groups of diffeomorphisms of the circle.

关于一般\(C^1\)族圆周微分同胚的旋转数不是赫尔德连续的

The Rotation Number for a Generic $C^1$ Family of Circle Diffeomorphisms is Not Hölder

研究圆周微分同胚单调单参数族中旋转数随参数的正则性，证明\(C^2\)单调族中最优赫尔德指数为\(1/2\)，\(C^{1+\alpha}\)族中旋转数正则性不超过\(\alpha/(1+\alpha)\)-赫尔德，得出一般\(C^1\)族旋转数非赫尔德连续。

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

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。

Quantitative statistical stability and linear response for irrational rotations and diffeomorphisms of the circle

We prove quantitative statistical stability results for a large class of small $C^{0}$ perturbations of circle diffeomorphisms with irrational rotation numbers. We show that if the rotation number is Diophantine the invariant measure varies in a Hölder way under perturbation of the map and the Hölder exponent depends on the Diophantine type of the rotation number. The set of admissable perturbations includes the ones coming from spatial discretization and hence numerical truncation. We also show linear response for smooth perturbations that preserve the rotation number, as well as for more gen…

One-parameter families of circle diffeomorphisms with strictly monotone rotation number

We show that if $f \colon S^1 \times S^1 \to S^1 \times S^1$ is $C^2$, with $f(x, t) = (f_t(x), t)$, and the rotation number of $f_t$ is equal to $t$ for all $t \in S^1$, then $f$ is topologically conjugate to the linear Dehn twist of the torus $(1&1 0&1)$. We prove a differentiability result where the assumption that the rotation number of $f_t$ is $t$ is weakened to say that the rotation number is strictly monotone in $t$.

Itineraries of rigid rotations and diffeomorphisms of the circle

We examine the itinerary of $0\in S^{1}=\R/\Z$ under the rotation by $α\in\R\bs\Q$. The motivating question is: if we are given only the itinerary of 0 relative to $I\subset S^{1}$, a finite union of closed intervals, can we recover $α$ and $I$? We prove that the itineraries do determine $α$ and $I$ up to certain equivalences. Then we present elementary methods for finding $α$ and $I$. Moreover, if $g:S^{1}\to S^{1}$ is a $C^{2}$, orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalenc…

Groups of Circle Diffeomorphisms

This book covers many of the recent results on group actions on the circle, with an emphasis in the differentiable case.

Cohomological equations and invariant distributions for minimal circle diffeomorphisms

Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As a consequence of this, we show that the space of real smooth coboundaries of such a diffeomorphism is closed if and only if its rotation number is Diophantine.

On conjugates and the asymptotic distortion of 1-dimensional $C^{1+bv}$ diffeomorphisms

We show that a $C^{1+bv}$ circle diffeomorphism with absolutely continuous derivative and irrational rotation number can be conjugated to diffeomorphisms that are $C^{1+bv}$ arbitrary close to the corresponding rotation. This improves a theorem of M.~Herman, who established the same result but starting with a $C^2$ diffeomorphism. We prove that the same holds for countable Abelian groups of circle diffeomorphisms acting freely, a result that is new even in the $C^{\infty}$ context. Related results and examples concerning the asymptotic distortion of diffeomorphisms are presented. Along this pa…

关于圆周微分同胚的随机游走与群作用的维数理论

Dimension theory of group actions by circle diffeomorphisms II: Minimal sets

本文研究圆周微分同胚随机游走与群作用的不变集维数，证明例外极小集的Hausdorff维数小于1，并在含重数为k+1的不动点时大于k/(k+1)，从而将Fuchs群经典结果推广到非线性情形。

Berger domains and Kolmogorov typicality of infinitely many invariant circles

Using the novel notion of parablender, P. Berger proved that the existence of finitely many attractors is not Kolmogorov typical in parametric families of diffeomorphisms. Here, motivated by the concept of Newhouse domains we define Berger domains for families of diffeomorphisms. As an application, we show that the coexistence of infinitely many attracting invariant smooth circles is Kolmogorov typical in certain non-sectionally dissipative Berger domains of parametric families in dimension three or greater.

交换圆微分同胚的同时重整化

Simultaneous renormalization of commuting circle diffeomorphisms

本文为交换圆微分同胚对定义重整化算子，在加速Brun算法驱动下，证明对有界型旋转向量指数快速收敛到刚性旋转，并推广到满测度类，同时证明算子拓扑双曲性。

Differentiable conjugacy for groups of area preserving circle diffeomorphisms

We study groups of circle diffeomorphisms whose action on the cylinder $\mathcal C=\mathbb S^1\times \mathbb S^1\setminus Δ$ preserves a volume form. We first show that such a group is topologically conjugate to a subgroup of $\rm{PSL}(2,\mathbb R)$, then discuss the existence of a differentiable conjugacy.

Distortion in Groups of Circle and Surface Diffeomorphisms

In these lectures we consider how algebraic properties of discrete subgroups of Lie groups restrict the possible actions of those groups on surfaces. The results show a strong parallel between the possible actions of such a group on the circle $S^1$ and the measure preserving actions on surfaces. Our aim is the study of the (non)-existence of actions of lattices in a large class of non-compact Lie groups on surfaces. A definitive analysis of the analogous question for actions on $S^1$ was carried out by É. Ghys. Our approach is topological and insofar as possible we try to isolate properties o…

Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function

We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the $C^1$-topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic group of finite index or a nonabelian free subgroup.

Maximal pseudometrics and distortion of circle diffeomorphisms

We initiate a study of distortion elements in the Polish groups $\mbox{Diff}_+^k(\mathbb{S}^1)$ ($1\leq k<\infty$), as well as $\mbox{Diff}_+^{1+AC}(\mathbb{S}^1)$, in terms of maximal metrics on these groups. We classify distortion in the $k=1$ case: a $C^1$ circle diffeomorphism is $C^1$-undistorted if and only if it has a hyperbolic periodic point. On the other hand, answering a question of Navas, we exhibit analytic circle diffeomorphisms with only non-hyperbolic fixed points which are $C^{1+AC}$-undistorted, and hence $C^k$-undistorted for all $k\geq 2$. In the appendix, we exhibit a maxi…

On embedding of arcs and circles in 3-manifolds in an application to dynamics of rough 3-diffeomorhisms with two-dimensional expanding attractor

A topological classification of many classes of dynamical systems with regular dynamics in low dimensions is often reduced to combinatorial invariants. In dimension 3 combinatorial invariants are proved to be insufficient even for simplest Morse-Smale diffeomorphisms. The complete topological invariant for the systems with a single saddle point on the 3-sphere is the embedding of the homotopy non-trivial knot into the manifold $\mathbb S^2\times\mathbb S^1$. If a diffeomorphism has several saddle points their unstable separatrices form arcs frames in the basin of the sink and circles frame in …

On the projective derivative cocycle for circle diffeomorphisms

We study the projective derivative as a cocycle of Möbius transformations over groups of circle diffeomorphisms. By computing precise expressions for this cocycle, we obtain several results about reducibility and almost reducibility to a cocycle of rotations. We also introduce an extension of this cocycle to the diagonal action on the 3-torus for which we generalize the previous results.

Diffeomorphisms with an anomalous growth of the differential

The paper is withdrawn.

The symplectic structure of renormalisation of circle diffeomorphisms with breaks

In this article we prove that iterated renormalisations of $\mathcal{C}^r$ circle diffeomorphisms with $d$ breaks, $r>2$, with given size of breaks, converge to an invariant family of piecewise Moebius maps, of dimension $2d$. We prove that this invariant family identifies with a \textit{relative character variety} $χ(π_1 Σ, \mathrm{PSL}(2,\mathbb{R}), \mathbf{h})$ where $Σ$ is a $d$-holed torus, and that the renormalisation operator identifies with a sub-action of the mapping class group $\mathrm{MCG}(Σ)$. This action is known to preserves a symplectic form, thanks to the work of Guruprasad-H…

A growth gap for diffeomorphisms of the interval

Given an orientation-preserving diffeomorphism of the interval [0;1], consider the uniform norm of the differential of its n-th iteration. We get a function of n called the growth sequence. Its asymptotic behaviour is an interesting invariant which naturally appears both in geometry of the diffeomorphisms groups and in smooth dynamics. Our main result is the following Gap Theorem: the growth rate of this sequence is either exponential, or at most quadratic with n. Further, we construct diffeomorphisms with quite an irregular behaviour of the growth sequence. This construction easily extends to…

Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms

We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.

Slow Diffeomorphisms of a Manifold with Two Dimensions Torus Action

The uniform norm of the differential of the n-th iteration of a diffeomorphism is called the growth sequence of the diffeomorphism. In this paper we show that there is no lower universal growth bound for volume preserving diffeomorphisms on manifolds with an effective two dimensions torus action by constructing a set of volume-preserving diffeomorphisms with arbitrarily slow growth.

Polynomial growth of the derivative for diffeomorphisms on tori

We consider area--preserving diffeomorphisms on tori with zero entropy. We classify ergodic area--preserving diffeomorphisms of the 3--torus for which the sequence $\{Df^n\}_{n\in{\Bbb N}}$ has polynomial growth. Roughly speaking, the main theorem says that every ergodic area--preserving $C^2$--diffeomorphism with polynomial uniform growth of the derivative is $C^2$--conjugate to a 2--steps skew product of the form \[\tor^3\ni(x_1,x_2,x_3)\mapsto (x_1+α,\ep x_2+β(x_1),x_3+γ(x_1,x_2))\in\tor^3,\] where $\ep=\pm 1$. We also indicate why there is no 4--dimensional analogue of the above result. Ra…

乒乓划分与实解析圆微分同胚的局部离散群，II：应用

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, II: Applications

该研究作为系列工作第二部分，基于第一部分提出的实解析圆微分同胚局部离散子群拓扑分类方法，证明了Dippolito关于不变康托集作用半共轭于分段线性作用的猜想，并构造了非富克斯型的局部离散极小作用实例。

Slow area-preserving diffeomorphisms of the torus

We construct area-preserving real analytic diffeomorphisms of the torus with unbounded growth sequences of arbitrarily slow growth.

乒乓划分与实解析圆周微分同胚的局部离散群，I：构造

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, I: Construction

本文针对实解析圆周微分同胚群的局部离散几乎自由子群，通过推广乒乓引理给出拓扑动力学分类方法，并在受限情形下解决了Dippolito的叶状结构理论古老猜想，相关推论将在配套论文详述。

A sharp smoothness of the conjugation of class P-homeomorphisms to diffeomorphisms

Let f be a class P -homeomorphism of the circle. We prove that there exists a piecewise analytic homeomorphism that conjugate f to a one-class P with prescribed break points lying on pairwise distinct orbits. As a consequence, we give a sharp estimate for the smoothness of a conjugation of class P -homeomorphism f of the circle satisfying the (D)-property (i.e. the product of f-jumps in the break points contained in a same orbit is trivial), to diffeomorphism. When f does not satisfy the (D)-property the conjugating homeomorphism is never piecewise C^1 and even more it is not absolutely contin…

关于不变圆周对分界线的累积

On the accumulation of separatrices by invariant circles

本文研究了辛微分同胚中分界线被不变圆周累积的条件，证明了辛自治向量场时间1映射的扰动下分界线会被正测度不变圆周集合累积，并给出了不被累积的反例。

A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms

Let M be a compact manifold of dimension at least 2, Diff^r(M) be the space of C^r diffeomorphisms of M. Define for any diffeomorphism f in Diff^r(M) number of isolated periodic points of period n by P_n(f)=# {isolated x in M: f^n(x)=x}. Artin--Mazur proved that for a dense set of diffeomorphisms the number of periodic points P_n(f) growth at most exponentially fast in n. The author proved that there is an open set N \subset Diff^r(M) such that for a Baire generic set of diffeomorphisms the number of periodic points P_n(f) growth arbitrarily fast. Arnold posed a problem: Prove that diffeomorph…

Abundance of fast growth of the number of periodic points in 2-dimensional area-preserving dynamics

We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there exists an open subset of the set of smooth area-preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points.

具有圈上断点的传递局部微分同胚的相变

Phase transitions for transitive local diffeomorphism with break points on the circle and Holder continuous potentials

本文研究了具有圈上断点的传递局部微分同胚在Hölder连续势函数下的相变行为，证明了Hölder连续势函数最多有两个相变，并给出了谱间隙性质的稠密集及多标度分析应用。

圆周上局部微分同胚的热力学与谱相变

Thermodynamical and spectral phase transition for local diffeomorphisms in the circle

本文研究圆周上非一致扩张且不可逆的局部微分同胚，证明其关于几何势的拓扑压函数非解析，并刻画了转移算子谱隙性质在临界参数处发生唯一热力学相变的条件。

Complex rotation numbers: bubbles and their intersections

The construction of complex rotation numbers, due to V.Arnold, gives rise to a fractal-like set "bubbles" related to a circle diffeomorphism. "Bubbles" is a complex analogue to Arnold tongues. This article contains a survey of the known properties of bubbles, as well as a variety of open questions. In particular, we show that bubbles can intersect and self-intersect, and provide approximate pictures of bubbles for perturbations of Möbius circle diffeomorphisms.