Sourav Bhattacharya, Alexander Blokh
We describe all possible bimodal over-twist patterns. In particular, we give an algorithm allowing one to determine what the left endpoint of the over-rotation interval of a given bimodal map is. We then define a new class of polymodal interval maps called well behaved, and generalize the above results onto well behaved maps.
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We show that the connectedness of the set of parameters for which the over-rotation interval of a bimodal interval map is constant. In other words, the over-rotation interval is a monotone function of a bimodal interval map.
We give conditions that characterize the existence of an absolutely continuous invariant probability measure for a degree one $C^2$ endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that both boundaries of its rotation interval are irrational numbers. Those conditions are satisfied when the boundary points of the rotation interval belong to a Diophantine class. In particular they hold for Lebesgue almost every rotation interval. The measure obtained is a global physical measure, and it is hyperbolic.
Inspired by a twist maps theorem of Mather we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map $g$ to the $k$-fold cover. For each irrational in the interior of the rotation set the collection of the $k$-fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$-dimensional ball in the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool is a generalization of a construction of Hedlund and Morse which …
We apply set-valued numerical methods to compute an accurate enclosure of the rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points, which is used to check the rationality of the rotation number. A few numerical experiments are presented to show that the implementation of interval methods produces a good enclosure of the rotation number of a circle map.
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.
Consider the two-parameter family of real analytic maps $F_{a,b}:x \mapsto x+ a+{b\over 2π} \sin(2πx)$ which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed interval $I$, the set of maps $F_{a,b}$ whose rotation interval is $I$, form a contractible set.
We prove that a periodic orbit $P$ with coprime over-rotation pair is an over-twist periodic orbit iff the $P$-linear map has the over-rotation interval with left endpoint equal to the over-rotation number of $P$. We then show that this result fails if the over-rotation pair of $P$ is not coprime. Examples of patterns with non-coprime over-rotation pairs are given so that these patterns have no block structure over over-twists but have over-rotation number equal to the left endpoint of the forced over-rotation interval (such patterns are called \emph{very badly ordered}). This presents a situa…
We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$-robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is uniformly large but the entropy approaches zero as much as desired. The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.
Let $Homeo\_0 (R 2 ; 0)$ be the set of all homeomorphisms of the plane isotopic to the identity and which fix 0. Recently in the article entitled "L'ensemble de rotation local autour d'un point fixe" Fr{é}d{é}ric Le Roux gave the definition of the local rotation set of an isotopy in $Homeo\_0 (R 2 ; 0)$ from the identity to a homeomorphism f and he asked if this set is always an interval. In this article we give a positive answers to this question and to the analogous question in the case of the open annulus.
We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $f_{\p}$ are parametrized by a quintuple $\p$ of real numbers satisfying inequations. Viewing $f_{\p}$ as a circle map, we show that it has a rotation number $ρ(f_{\p})$ and we compute $ρ(f_{\p})$ as a function of $\p$ in terms of Hecke-Mahler series. As a corollary, we prove that $ρ(f_{\p})$ is a rational number when the components of $\p$ are algebraic numbers.
A disjoint rotation map is an interval exchange transformation (IET) on the unit interval that acts by rotation on a finite number of invariant subintervals. It is currently unknown whether the group E of all IETs possesses any non-abelian free subgroups. It is shown that it is not possible for a disjoint rotation map to occur in a subgroup of E that is isomorphic to a non-abelian free group.
We introduce {\it twist unimodal maps} of the interval and describe their structure. Sufficient conditions for the growth of over-rotation interval in families of maps are given.
Interval maps mimicking circle rotations
该研究聚焦一类双参数区间映射的动力学，证明当参数满足特定有理条件且另一参数足够大时存在吸引几乎所有轨迹的周期轨道，并通过数值发现其他现象给出解释。
In this article we present an efficient algorithm to compute rotation intervals of circle maps of degree one. It is based on the computation of the rotation number of a monotone circle map of degree one with a constant section. The main strength of this algorithm is that it computes \emph{exactly} the rotation interval of a natural subclass of the continuous non-invertible degree one circle maps. We also compare our algorithm with other existing ones by plotting the Devil's Staircase of a one-parameter family of maps and the Arnold Tongues and rotation intervals of some special non-differentia…
We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok-Stepin, Veech, and Avila-Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have…
Interval translation maps (ITMs) are a non-invertible generalization of interval exchange transformations (IETs). The dynamics of finite type ITMs is similar to IETs, while infinite type ITMs are known to exhibit new interesting effects. In this paper, we prove the finiteness conjecture for the ITMs of three intervals. Namely, the subset of ITMs of finite type contains an open, dense, and full Lebesgue measure subset of the space of ITMs of three intervals. For this, we show that any ITM of three intervals can be reduced either to a rotation or to a double rotation.
We study the ergodic properties of compositions of interval exchange transformations and rotations. We show that for any interval exchange transformation T, there is a full measure set of αin [0, 1) so that T composed with R_α is uniquely ergodic, where R_α is rotation by α.
This paper presents quantitative shrinking target results for rotations and interval exchange transformations. To do this a quantitative version of a unique ergodicity criterion of Boshernitzan is established.
We show that given $n$ normalized intervals on the unit circle, the numbers of visits of $d$ random rotations to these intervals have a joint limiting distribution as lengths of trajectories tend to infinity. If $d$ then tends to infinity, then the numbers of points in different intervals become asymptotically independent unless an arithmetic obstruction arises. This is a generalization of earlier results of J. Marklof.
For a continuous map on a topological graph containing a unique loop S it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and for every rational r in this interval there exists a periodic point of rotation number r. The whole rotation set (i.e. the set of all rotation numbers) may not be connected and it is not known in general whether it is closed. The graph sigma is the space consisting in an interval attached by one of its endpoints to a circle. We show that, for a map of degree …
We show that the coding of rotation by $α$ on $m$ intervals with rationally independent lengths can be recoded over $m$ Sturmian words of angle $α.$ More precisely, for a given $m$ an universal automaton is constructed such that the edge indexed by the vector of values of the $i$th letter on each Sturmian word gives the value of the $i$th letter of the coding of rotation.
We consider a prototypical two-parameter family of invertible maps of $\mathbb{Z}^2$, representing rotations with decreasing rotation number. These maps describe the dynamics inside the island chains of a piecewise affine discrete twist map of the torus, in the limit of fine discretisation. We prove that there is a set of full density of points which, depending of the parameter values, are either periodic or escape to infinity. The proof is based on the analysis of an interval-exchange map over the integers, with infinitely many intervals.
Graph maps that are homotopic to the identity and that permute the vertices are studied. Given a periodic point for such a map, a {\em rotation element} is defined in terms of the fundamental group. A number of results are proved about the rotation elements associated to periodic points in a given edge of the graph. Most of the results show that the existence of two periodic points with certain rotation elements will imply an infinite family of other periodic points with related rotation elements. These results for periodic points can be considered as generalizations of the rotation interval f…
Roth type irrational rotation numbers have several equivalent arithmetical characterizations as well as several equivalent characterizations in terms of the dynamics of the corresponding circle rotations. In this paper we investigate how to generalize Roth-like Diophantine conditions to interval exchange maps. If one considers the dynamics in parameter space one can introduce two nonequivalent Roth-type conditions, the first (condition (Z)) by means of the Zorich cocyle, the second (condition (A)) by means of a further acceleration of the continued fraction algorithm introduced in [10]. A thir…
Forcing Minimal Interval Patterns as Interval Exchange Transformations
本研究证明过扭转模式共轭于等距分段数有界的区间交换变换（限制在单周期上），且该界仅依赖模态数，与周期、过旋转数无关，为区间模式与区间交换变换的关联提供了新结论。
Let $0<λ<1$. We consider the one-parameter family of circle $λ$-affine contractions $f_δ:x \in [0,1) \mapsto λx + δ\; {\rm mod}\,1 $, where $0 \le δ<1$. Let $ρ$ be the rotation number of the map $f_δ$. We will give some numerical relations between the values of $λ,δ$ and $ρ$, essentially using Hecke-Mahler series and a tree structure. When both parameters $λ$ and $δ$ are algebraic numbers, we show that $ρ$ is a rational number. Moreover, in the case $λ$ and $δ$ are rational, we give an explicit upper bound for the height of $ρ$ under an assumption on $λ$.
For a continuous map on a topological graph containing a unique loop S, it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and for every rational r in this interval there exists a periodic point of rotation number r. The whole rotation set (i.e. the set of all rotation numbers) may not be connected and it is not known in general whether it is closed. A sun graph is the space consisting in finitely many segments attached by one of their endpoints to a circle. We show that, for a map …
Denote by $G$ the group of interval exchange transformations (IETs) on the unit interval. Let $G_{per}\subset G$ be the subgroup generated by torsion elements in $G$ (periodic IETs), and let $G_{rot}\subset G$ be the subset of 2-IETs (rotations). The elements of the subgroup $G_1=< G_{per},G_{rot}>\subset G$ (generated by the sets $G_{per}$ and $G_{rot}$) are characterized constructively in terms of their Sah-Arnoux-Fathi (SAF) invariant. The characterization implies that a non-rotation type 3-IET lies in $G_1$ if and only if the lengths of its exchanged intervals are linearly dependent over $…
For a continuous map on a topological graph containing a loop $S$ it is possible to define the degree (with respect to the loop $S$) and, for a map of degree $1$, rotation numbers. We study the rotation set of these maps and the periods of periodic points having a given rotation number. We show that, if the graph has a single loop $S$ then the set of rotation numbers of points in $S$ has some properties similar to the rotation set of a circle map; in particular it is a compact interval and for every rational $α$ in this interval there exists a periodic point of rotation number $α$. For a speci…
We are concerned with describing the structure of the set of points in the unit interval which, when subjected to rotation by irrational alpha modulo one, for all finite portions of the orbit contain at least as many points in the bottom half of the interval as in the top half. Specifically, an inductive procedure for describing the set based on the continued fraction expansion of alpha is developed, leading into a discussion of the Hausdorff dimension of this set. Depending on the parameter alpha, all possible dimensions may be achieved, and the essential infimum (with respect to alpha) of th…
Rotation number and dynamics of 3-interval piecewise $λ$-affine contractions
本文研究3-区间分段$\lambda$-仿射压缩映射族，给定$\lambda$和$d$，确定参数$\delta$和$a$使映射具有指定旋转数，并保证其符号动力学包含相应旋转的符号动力学，从而刻画周期轨道数量与Cantor集吸引子的复杂度。
We look at $d$-point extensions of a rotation of angle $α$ with $r$ marked points, generalizing the examples of Veech 1969 and Sataev 1975, together with the square-tiled interval exchange transformations of \cite{fh2}. We study the property of rigidity, in function of the Ostrowski expansions of the marked points by $α$: we prove that $T$ is rigid when $α$ has unbounded partial quotients, and that $T$ is not rigid when the natural coding of the underlying rotation with marked points is linearly recurrent. But there remains an interesting grey zone between these two cases, in which we have onl…
Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to $r$-ball of the initial point is bounded below. In this paper, we consider how the bounded type condition of irrational is generalized into interval exchange maps.
The twist interval of a twist map on the annulus $A=\mathbb{T}\times [0,1]$ has nonempty interior if $f$ preserves the area, but could be degenerate for general twist maps. In this note, we show that if a twist map $f$ is non-wandering, then the twist interval of $f$ is non-degenerate. Moreover, if there are two disjoint invariant curves of $f$, then their rotation numbers must be different (no matter if they are rational or irrational).
Twist like behavior in non-twist patterns of triods
本文证明了三脚架上模式旋转数与强制旋转区间端点重合的充分条件，发现了非扭转且无块结构的“奇怪有序”模式，该模式通过分段单调映射半共轭于圆旋转，并提供了构造任意旋转对单峰模式的算法。
Let $f : [0,1)\rightarrow [0,1)$ be a $2$-interval piecewise affine increasing map which is injective but not surjective. Such a map $f$ has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of $f$ thanks to two specific functions $δ$ and $ϕ$ depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of $f$ is rational, when the three parameters are algebraic numbers.
We study C^2 weakly order preserving circle maps with a flat interval. In particular we are interested in the geometry of the mapping near to the singularities at the boundary of the flat interval. Without any assumption on the rotation number we show that the geometry is degenerate when the degree of the singularities is less than or equal to two and becomes bounded when the degree goes to three. As an example of application, the result is applied to study Cherry flows.
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。
Upper Bound for Permanent Saturation of Metric Graphs using Interval Exchange Transformations
通过区间交换变换研究有限度量图中永久ε-饱和时间的上界，证明了当图包含两条长度比为无理数的闭路时，饱和时间存在显式上界，有界型旋转给出ε^{-1}阶上界。
In this paper we study quasi-symmetric conjugations of $C^2$ weakly order-preserving circle maps with a flat interval. Under the assumption that the maps have the same rotation number of bounded type and that bounded geometry holds we construct a quasi-symmetric conjugation between their non-wandering sets. Further, this conjugation is extended to a quasi-symmetric circle homeomorphism. Our proof techniques hinge on real-dynamic methods allowing us to construct the conjugation under general and natural assumptions.
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