我们给出了以无理圆周旋转为拓扑因子的二维环面上的无周期点同胚的完整刻画。给定一个无周期点且在某个有理方向上具有一致有界旋转偏差的二维环面同胚，我们证明了环状性及其非游荡集的几何结构是存在无理圆周旋转作为拓扑因子的唯一可能障碍。通过对诱导的ρ-中心化斜积动力学的非常精确的研究，我们大幅扩展并推广了T. Jäger之前的成果[Inventiones Mathematicae, 176 (2009), n. 3, 601-616]。
英文摘要：
We provide a complete characterization of periodic point free homeomorphisms of the $2$-torus admitting irrational circle rotations as topological factors. Given a homeomorphism of the $2$-torus without periodic points and exhibiting uniformly bounded rotational deviations with respect to a rational direction, we show that annularity and the geometry of its non-wandering set are the only possible obstructions for the existence of an irrational circle rotation as topological factor. Through a very precise study of the dynamics of the induced $ρ$-centralized skew-product, we extend and generalize considerably previous results of T. Jäger [Inventiones Mathematicae, 176 (2009), n. 3, 601-616].
We provide an equivalent characterisation for the existence of one-dimensional irrational rotation factors of conservative torus homeomorphisms that are not eventually annular. It states that an area-preserving non-annular torus homeomorphism $f$ is semiconjugate to an irrational rotation $R_α$ of the circle if and only if there exists a well-defined speed of rotation in some rational direction on the torus, and the deviations from the constant rotation in this direction are uniformly bounded. By means of a counterexample, we also demonstrate that a similar characterisation does not hold for e…
In the context of the Franks-Misiurewicz Conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterise these maps by the existence of an invariant foliation by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalising a well-known res…
A conservative irrational pseudo-rotation of the two-torus is semi-conjugate to the irrational rotation if and only if it has the property of bounded mean motion [10]. (Here 'irrational pseudo-rotation' means a toral homeomorphism with uniquely defined, totally irrational rotation vector.) The aim of this note is to explore this concept some further. For instance, we provide an example which shows that the preceding statement does not hold in the non-conservative case. Further, we collect a number of observations concerning the case where the bounded mean motion property fails. In particular, …
Mary Rees has constructed a minimal homeomorphism of the 2-torus with positive topological entropy. This homeomorphism f is obtained by enriching the dynamics of an irrational rotation R. We improve Rees construction, allowing to start with any homeomorphism R instead of an irrational rotation and to control precisely the measurable dynamics of f. This yields in particular the following result: Any compact manifold of dimension d>1 which carries a minimal uniquely ergodic homeomorphism also carries a minimal uniquely ergodic homeomorphism with positive topological entropy. More generally, give…
Let $G$ be the group ${\rm PAff}_+({\bf S}^1)$ of piecewise--affine circle homeomorphisms or the group ${\Diff}^{\infty}(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in $G$ is given.
For any irrational number $α$, there exists an ergodic area preserving homeomorphism of the closed annulus which is isotopic to the identitity, admits no compact invariant set contained in the interior of the annulus, and has the rotation number $α$.
We show that a toral homeomorphism which is homotopic to the identity and topologically semiconjugate to an irrational rotation of the circle is always a pseudo-rotation (i.e. its rotation set is a single point). In combination with recent results, this allows us to complete the study of the Franks-Misiurewicz conjecture in the minimal case.
We prove that a homeomorphism of the torus homotopic to the identity whose rotation set is reduced to a single totally irrational vector is chain-recurrent. In fact, we show that pseudo-orbits can be chosen with a small number of jumps, in particular, that the nonwandering set is weakly transitive. We give an example showing that the nonwandering set of such a homeomorphism may not be transitive.
We give an equivalent characterisation for the existence of a semi-conjugacy to an irrational rotation for conservative homeomorphisms of the two-torus. This leads to an analogue of Poincare's classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property. For minimal toral homeomorphisms, the result extends to arbitrary dimensions. Further, we provide a basic classification for the dynamics of non-wandering toral homeomorphisms homotopic to the identitiy.
As was known to H. Poincare, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms; that is, torus homeomorphisms isotopic to the identity for which the rotation set is a point with rationally independent irrational coordinates.
In this paper we prove the $C^1$-density of every $C^r$-conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number.
We prove that any minimal $2$-torus homeomorphism which is isotopic to the identity and whose rotation set is not just a point exhibits uniformly bounded rotational deviations on the perpendicular direction to the rotation set. As a consequence of this, we show that any such homeomorphism is topologically mixing and we prove Franks-Misiurewicz conjecture under the assumption of minimality.
Despite a general lack of a useful 3 dimensional prime end theory, if h is a homeomorphism of the product of the real line R and the closure of a bounded open contractible planar set U, then h typically admits a notion of rotation number analogous to the usual one.
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.
Given an irrational rotation $T$ on $\M T$ we settle necessary and sufficient conditions on a step function $ϕ$ and $t\in \M T$ for the existence of measurable solutions to the cohomogical equation $$\exp{(2iπϕ)}=\e{2iπt}f/f\rond T.$$ This yields a characterization of eigenvalues and eigenfunctions for several transformations arising from irrational rotations and step functions: cylinder flows, special flows, induced maps... From there we give constructions of special flows and three-interval exchange transformations with unusual spectral properties. In both cases we exhibit examples with Kron…
Rotation vectors, as defined for homeomorphisms of the torus that are isotopic to the identity, are generalized to such homeomorphisms of any complete Riemannian manifold with non-positive sectional curvature. These generalized rotation vectors are shown to exist for almost every orbit of such a dynamical system with respect to any invariant measure with compact support. The concept is then extended to flows and, as an application, it is shown how non-null rotation vectors can be used to construct a measurable semi-conjugacy between a given flow and the geodesic flow of a manifold.
Let $K_1$, $K_2$ $\subset$ $R^2$ be two convex, compact sets. We would like to know if there are commuting torus homeomorphisms $f$ and $h$ homotopic to the identity, with lifts $\tilde f$ and $\tilde h$, such that $K_1$ and $K_2$ are their rotation sets, respectively. In this work, we proof some cases where it cannot happen, assuming some restrictions on rotation sets.
We show that if the rotation set of a homeomorphism of the torus is stable under small perturbations of the dynamics, then it is a convex polygon with rational vertices. We also show that such homeomorphisms are $C^0$-generic and have bounded rotational deviations (even for pseudo-orbits). The results hold both in the area-preserving setting and in the general setting. When the rotation set is stable, we give explicit estimates on the type of rationals that may appear as vertices of rotation sets in terms of the stability constants.
Given an integer m>1 we consider Zm-equivariant and orientation preserving homeomorphisms in the plane with an asymptotically stable fixed point at the origin. We present examples without periodic points and having some complicated dynamical features. The key is a preliminary construction of Zm-equivariant Denjoy maps of the circle.
We prove that for a torus homeomorphism isotopic to the identity and with a lift whose rotation set is an interval, either every rational point in the rotation set is realized by a periodic orbit, or there exists an annular, essential, periodic set. In the latter case we give a qualitative description of the dynamics.
We prove the existence of an open and dense set D\subset? Homeo0(T2) (set of toral homeomorphisms homotopic to the identity) such that the rotation set of any element in D is a rational polygon. We also extend this result to the set of axiom A dif- feomorphisms in Homeo0(T2). Further we observe the existence of minimal sets whose rotation set is a non-trivial segment, for an open set in Homeo0(T2).
This article deals with directional rotational deviations for non-wandering periodic point free homeomorphisms of the 2-torus which are homotopic to the identity. We prove that under mild assumptions, such a homeomorphism exhibits uniformly bounded rotational deviations in some direction if and only it leaves invariant a pseudo-foliation, a notion which is a slight generalization of classical one-dimensional foliations. To get these results, we introduce a novel object called $\tildeρ$-centralized skew-product and their associated stable sets at infinity.
We show that, if $f$ is a homeomorphism of the 2--torus isotopic to the identity, and its lift $\widetilde f$ is transitive, or even if it is transitive outside of the lift of the elliptic islands, then $(0,0)$ is in the interior of the rotation set of $\widetilde f.$ This proves a particular case of Boyland's conjecture.
We prove that the Calabi invariant of a $C^1$ pseudo-rotation of the unit disk, that coincides with a rotation on the unit circle, is equal to its rotation number. This result has been shown some years ago by Michael Hutchings (under very slightly stronger hypothesis). While the original proof used Embedded Contact Homology techniques, the proof of this article uses generating functions and the dynamics of the induced gradient flow.
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…
The goals of this paper are to obtain theoretical models of what happens when a computer calculates the rotation set of a homeomorphism, and to find a good algorithm to perform simulations of this rotation set. To do that we introduce the notion of observable rotation set, which takes into account the fact that we can only detect phenomenon appearing on positive Lebesgue measure sets; we also define the asymptotic discretized rotation set which in addition takes into account the fact that the computer calculates with a finite number of digits It appears that both theoretical results and simula…
The first result of the paper (Theorem 1.1) is an explicit construction of unimodal maps that are semiconjugate, on the post-critical set, to the circle rotation by an arbitrary irrational angle $θ\in(3/5,2/3)$. Our construction is a generalization of the construction by Milnor and Lyubich [LM] of the Fibonacci unimodal maps semi-conjugate to the circle rotation by the golden ratio. Generalizing a theorem by Milnor and Lyubich for the Fibonacci map, we prove that the Hausdorff dimension of the post-critical set of our unimodal maps is $0$, provided the denominators of the continued fraction of…
We show that for group actions on locally connected spaces the maximal equicontinuous factor map is always monotone, that is, the preimages of single points are connected. As an application, we obtain that if the maximal continuous factor of a homeomorphism of the two-torus is minimal, then it is either (i) an irrational translation of the two-torus, (ii) an irrational rotation on the circle or (iii) the identity on a singleton.
For each {\it well approximable} irrational $θ$, we provide an explicit rank-one construction of the $e^{2πiθ}$-rotation $R_θ$ on the circle $\Bbb T$. This solves "almost surely" a problem by del Junco. For {\it every} irrational $θ$, we construct explicitly a rank-one transformation with an eigenvalue $e^{2πiθ}$. For every irrational $θ$, two infinite $σ$-finite invariant measures $μ_θ$ and $μ_θ'$ on $\Bbb T$ are constructed explicitly such that $(\Bbb T,μ_θ, R_θ)$ is {\it rigid} and of rank one and $(\Bbb T,μ_θ', R_θ)$ is of {\it zero type} and of rank one. The centralizer of the latter syst…
Consider a homeomorphism h of the closed annulus S^1*[0,1], isotopic to the identity, such that the rotation set of h is reduced to a single irrational number alpha (we say that h is an irrational pseudo-rotation). For every positive integer n, we prove that there exists a simple arc gamma joining one of the boundary component of the annulus to the other one, such that gamma is disjoint from its n first iterates under h. As a corollary, we obtain that the rigid rotation of angle alpha can be approximated by homeomorphisms conjugate to h. The first result stated above is an analog of a theorem …
Let f be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector (p/q,r/q), then f has a topologically monotone periodic orbit with the same rotation vector.
Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a lift of $f$ to the universal cover of $A$, one defines the rotation set $\tilde ρ(X)$ of $X$ by means of the invariant probabilities on $X$, as well as the prime end rotation number $\checkρ_\pm$ of $U_\pm$. The purpose of this paper is to show that $\checkρ_\pm$ belongs to $\tildeρ(X)$ for any separating invariant continuum $X$.
Let $f_{i},i=1,2$ be orientation preserving circle homeomorphisms with a finite number of break points, at which the first derivatives $Df_{i}$ have jumps, and with identical irrational rotation number $ρ=ρ_{f_{1}}=ρ_{f_{2}}.$ The jump ratio of $f_{i}$ at the break point $b$ is denoted by $σ_{f_{i}}(b)$, i.e. $σ_{f_{i}}(b):=\frac{Df_{i}(b-0)}{Df_{i}(b+0)}$. Denote by $σ_{f_{i}}, i=1,2,$ the total jump ratio given by the product over all break points $b$ of the jump ratios $σ_{f_{i}}(b)$ of $f_{i}$. We prove, that for circle homeomorphisms $f_{i}, i=1,2$, which are $C^{2+\varepsilon}, \varepsil…
Let S be a compact, oriented surface with negative Euler characteristic and let f be a homeomorphism of S that is isotopic to the identity. If there exists a periodic orbit with a non-zero rotation vector, then there exists a simple braid with the same rotation vector.
This paper studies homeomorphisms of the closed annulus that are isotopic to the identity from the viewpoint of rotation theory, using a newly developed forcing theory for surface homeomorphisms. Our first result is a solution to the so called strong form of Boyland's Conjecture on the closed annulus: Assume $f$ is a homeomorphism of $\overline{\mathbb{A}}:=(\mathbb{R}/\mathbb{Z})\times [0,1]$ which is isotopic to the identity and preserves a Borel probability measure $μ$ with full support. We prove that if the rotation set of $f$ is a non-trivial segment, then the rotation number of the measu…
It is proved that all special flows over the rotation by an irrational $α$ with bounded partial quotients and under $f$ which is piecewise absolutely continuous with a non-zero sum of jumps are mildly mixing. Such flows are also shown to enjoy a condition which emulates the Ratner condition introduced in \cite{Rat}. As a consequence we construct a smooth vector--field on $\T^2$ with one singularity point such that the corresponding flow $(ϕ_t)_{t\in\R}$ preserves a smooth measure, its set of ergodic components consists of a family of periodic orbits and one component of positive measure on whi…
We consider the rotation set $ρ(F)$ for a lift $F$ of an area preserving homeomorphism $f: \t^2\to \t^2$, which is homotopic to the identity. The relationship between this set and the existence of periodic points for $f$ is least well understood in the case when this set is a line segment. We show that in this case if a vector $v$ lies in $ρ(F)$ and has both co-ordinates rational, then there is a periodic point $x\in \t^2$ with the property that $$\frac{F^q(x_0)-x_0}q = v$$ where $x_0\in \re^2$ is any lift of $x$ and $q$ is the least period of $x$.
We consider special flows over the rotation on the circle by an irrational $α$ under roof functions of bounded variation. The roof functions, in the Lebesgue decomposition, are assumed to have a continuous singular part coming from a quasi-similar Cantor set (including the Devil's staircase case). Moreover, a finite number of discontinuities is allowed. Assuming that $α$ has bounded partial quotients, we prove that all such flows are weakly mixing and enjoy weak Ratner's property. Moreover, we provide a sufficient condition for the roof function to obtain a stability of the cocycle Ratner's pr…
We consider special flows over the rotation by an irrational $α$ under the roof functions of bounded variation without continuous, singular part in the Lebesgue decomposition and the sum of jumps $\neq 0$. We show that all such flows are weakly mixing. Under the additional assumption that $α$ has bounded partial quotients, we study weak Ratner's property. We establish this property whenever an additional condition (stable under sufficiently small perturbations) on the set of jumps is satisfied. While it is classical that the flows under consideration are not mixing, one more condition on the s…
It is well known that the rotation number of a circle homeomorphism defined by H. Poincaré allows to completely understand the dynamics of such a map from the topological point of view. In this paper, we collect some results concerning the existence of topological dynamical models associated to rotation sets of homeomorphisms of $\T^2$.
We extend Poincaré's theory of orientation-preserving homeomorphisms from the circle to circloids with decomposable boundary. As special cases, this includes both decomposable cofrontiers and decomposable cobasin boundaries. More precisely, we show that if the rotation number on an invariant circloid $A$ of a surface homeomorphism is irrational and the boundary of $A$ is decomposable, then the dynamics are monotonically semiconjugate to the respective irrational rotation. This complements classical results by Barge and Gillette on the equivalence between rational rotation numbers and the exist…
Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a lift of $f$ to the universal cover of $A$, one defines the rotation set $\tilde ρ(X)$ of $X$ by means of the invariant probabilities on $X$. For any rational number $p/q\in \tildeρ(X)$, $f$ is shown to admit a $p/q$ periodic point in $X$, provided that (1) $X$ consists of nonwandering points or (2) $X$ is an attractor and the frontiers of $U_-$ and $U_+$ coinci…
We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. I…
We study a linear cocycle over irrational rotation $σ_ω(x) = x + ω$ of a circle $\mathbb{T}^{1}$. It is supposed the cocycle is generated by a $C^{1}$-map $A_{\varepsilon}: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which depends on a small parameter $\varepsilon\ll 1$ and has the form of the Poincaré map corresponding to a singularly perturbed Schrödinger equation. Under assumption the eigenvalues of $A_{\varepsilon}(x)$ to be of the form $\exp(\pm λ(x)/\varepsilon)$, where $λ(x)$ is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect t…
Poincare's classification of the dynamics of homeomorphisms of the circle is one of the earliest, but still one of the most elegant, classification results in dynamical systems. Here we generalize this to quasiperiodically forced circle homeomorphisms, which have been the subject of considerable interest in recent years. Herman already showed two decades ago that a unique rotation number exists for all orbits in the quasiperiodically forced case. However, unlike the unforced case, no a priori bounds exist for the deviations from the average rotation. This plays an important role in the attempt…
We construct a family $\{Φ_t\}_{t\in[0,1]}$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $ρ(Φ_t)$ can be described explicitly. We analyze the bifurcations and typical behavior of rotation sets in the family, providing insight into the general questions of toral rotation set bifurcations and prevalence. We show that there is a full measure subset of $[0,1]$, consisting of infinitely many mutually disjoint non-trivial closed intervals, on each of which the rotation set mode locks to a constant polygon with rational vertices; that the generic rot…
Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_0$ with Lefschetz index one. Le Roux conjectured that $z_0$ is accumulated by periodic orbits. In this article, we will approach Le Roux's conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_0$ and if the area of $M$ is finite, then $z_0$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_0$ is the limit in weak-star topology of a sequence of invaria…
We consider skew product extension of irrational rotations on the circle by $\Z^2$ determined by an integer valued function as well as a fixed point on the circle. We study ergodic components of such extension.
We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically nontrivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincaré-Birkhoff Theorem for torus homeomorphisms: in the abse…
