Irrational Rotation Number Research Articles

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

Abstract We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

Uniqueness of Denjoy minimal sets for twist maps with zero entropy *

For monotone twist maps with zero topological entropy, we show that the set of recurrent points with irrational rotation number α can be described by a single orientation preserving circle homeomorphism and hence there is either an invariant circle, or a unique Denjoy minimal set, of rotation number α.

Analytic maps of parabolic and elliptic type with trivial centralisers

We prove that for a dense set of irrational numbers \alpha , the analytic centraliser of the map e^{2\pi i \alpha} z+ z^2 near 0 is trivial. We also prove that some analytic circle diffeomorphisms in the Arnol’d family, with irrational rotation numbers, have trivial centralisers. These provide the first examples of such maps with trivial centralisers.

Renormalization of Bicritical Circle Maps

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 of a single critical point. When combined with some recent papers, our main theorem implies $C^{1+\alpha}$ rigidity for real-analytic bicritical circle maps with rotation number of bounded type.

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

<p style='text-indent:20px;'>We prove quantitative statistical stability results for a large class of small <inline-formula><tex-math id="M1">\begin{document}$ C^{0} $\end{document}</tex-math></inline-formula> 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 admissible 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 general ones. This is done by means of classical tools from KAM theory, while the quantitative stability results are obtained by transfer operator techniques applied to suitable spaces of measures with a weak topology.</p>

On Conjugates and the Asymptotic Distortion of One-Dimensional C1+bv Diffeomorphisms

Abstract 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 path, we provide a straightened version of the classical Denjoy–Kocsma inequality for absolutely continuous potentials.$\hfill$In memory of Jean-Christophe Yoccoz

Principal Foliations of Surfaces near Ellipsoids

The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar\'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has \emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.

A Note on the Conjugacy Between Two Critical Circle Maps

We study a conjugacy between two critical circle homeomorphisms with irrational rotation number. Let fi, i = 1, 2 be a C3 circle homeomorphisms with critical point x(i) cr of the order 2mi + 1. We prove that if 2m1 + 1 ̸= 2m2 + 1, then conjugating between f1 and f2 is a singular function. Keywords: circle homeomorphism, critical point, conjugating map, rotation number, singular function

Prime ends dynamics in parametrised families of rotational attractors

We provide several new examples in dynamics on the $2$-sphere, with the emphasis on better understanding the induced boundary dynamics of invariant domains in parametrized families. First, motivated by a topological version of the Poincare-Bendixson Theorem obtained recently by Koropecki and Passeggi, we show the existence of homeomorphisms of $\mathbb{S}^2$ with Lakes of Wada rotational attractors, with an arbitrarily large number of complementary domains, and with or without fixed points, that are arbitrarily close to the identity. This answers a question of Le Roux. Second, from reduced Arnold's family we construct a parametrised family of Birkhoff-like cofrontier attractors, where at least for uncountably many choices of the parameters, two distinct irrational prime ends rotation numbers are induced from the two complementary domains. This example complements the resolution of Walker's Conjecture by Koropecki, Le Calvez and Nassiri from 2015. Third, answering a question of Boyland, we show that there exists a non-transitive Birkhoff-like attracting cofrontier which is obtained from a BBM embedding of inverse limit of circles, such that the interior prime ends rotation number belongs to the interior of the rotation interval of the cofrontier dynamics. There exists another BBM embedding of the same attractor so that the two induced prime ends rotation numbers are exactly the two endpoints of the rotation interval.

Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

We consider order-preserving $C^3$ circle maps with a flat piece, irrational rotation number and critical exponents $(l_1, l_2)$. We detect a change in the geometry of the system. For $(l_1, l_2) \in [1, 2]^2$ the geometry is degenerate and becomes bounded for $(l_1, l_2) \in [2, \infty)^2 \backslash \{(2, 2)\}$. When the rotation number is of the form $[abab \ldots]$; for some $a, b \in \mathbb{N}^*$, the geometry is bounded for $(l_1, l_2)$ belonging above a curve defined on $]1, +\infty[^2$. As a consequence, we estimate the Hausdorff dimension of the nonwandering set $K_f=\mathcal{S}^1\backslash \bigcup^\infty_{i=0}f^{-i}(U)$. Precisely, the Hausdorff dimension of this set is equal to zero when the geometry is degenerate and it is strictly positive when the geometry is bounded.

Rigidity of critical circle maps

We prove that any two C4 critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a C1 circle diffeomorphism. The conjugacy is C1+α for a full Lebesgue measure set of rotation numbers.

Continuity of depinning force

Monotone recurrence relations give rise to a class of dynamical systems on the high-dimensional cylinder which generalizes the class of monotone twist maps. A solution of a monotone recurrence relation corresponds to an equilibrium of the generalized Frenkel–Kontorova (FK) model. The Aubry–Mather theory for monotone recurrence relations says that for each ω∈R there is a Birkhoff minimizer with rotation number ω. Whether all Birkhoff minimizers of rotation number ω form a foliation is a question like whether there is an invariant circle with rotation number ω for a monotone twist map. In this paper we give a criterion for the existence of minimal foliations and study its continuity.The depinning force Fd(ω), depending on rotation numbers, is a critical value of the driving force for the FK model, under which there are Birkhoff equilibria and hence the system is pinned, and above which there are no Birkhoff equilibria of rotation number ω and the system is sliding. We show that all Birkhoff minimizers with irrational rotation number ω form a foliation if and only if Fd(ω)=0. If ω=p/q is rational, then Fd(p/q)=0 if and only if all (p,q)-periodic Birkhoff minimizers constitute a foliation. Moreover, we show that Fd(ω) is continuous at irrational ω and Hölder continuous at Diophantine points, and it also depends continuously on system parameters.

Type III representations and modular spectral triples for the noncommutative torus

It is well known that for any irrational rotation number α, the noncommutative torus Aα must have representations π such that the generated von Neumann algebra π(Aα)″ is of type III. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator.In the present paper, we show that this program can be carried out, at least when α is a Liouville number satisfying a faster approximation property by rationals. In this case, we exhibit several type II∞ and IIIλ, λ∈[0,1], factor representations and modular spectral triples.The method developed in the present paper can be generalised to CCR algebras based on a locally compact abelian group equipped with a symplectic form.

Robust local Hölder rigidity of circle maps with breaks

We prove that, for every ε∈(0,1), every two C2+α-smooth (α>0) circle diffeomorphisms with a break point, i.e. circle diffeomorphisms with a single singular point where the derivative has a jump discontinuity, with the same irrational rotation number ρ∈(0,1) and the same size of the break c∈R+\\{1}, are conjugate to each other via a conjugacy which is (1−ε)-Hölder continuous at the break points. An analogous result does not hold for circle diffeomorphisms even when they are analytic.

Denjoy minimal sets and Birkhoff periodic orbits for non-exact monotone twist maps

A non-exact monotone twist map φ¯F is a composition of an exact monotone twist map φ¯ with a generating function H and a vertical translation VF with VF((x,y))=(x,y−F). We show in this paper that for each ω∈R, there exists a critical value Fd(ω)≥0 depending on H and ω such that for 0≤F≤Fd(ω), the non-exact twist map φ¯F has an invariant Denjoy minimal set with irrational rotation number ω lying on a Lipschitz graph, or Birkhoff (p,q)-periodic orbits for rational ω=p/q. Like the Aubry–Mather theory, we also construct heteroclinic orbits connecting Birkhoff periodic orbits, and show that quasi-periodic orbits in these Denjoy minimal sets can be approximated by periodic orbits. In particular, we demonstrate that at the critical value F=Fd(ω), the Denjoy minimal set is not uniformly hyperbolic and can be approximated by smooth curves.

Real bounds and quasisymmetric rigidity of multicritical circle maps

Let f , g : S 1 → S 1 f, g:S^1\to S^1 be two C 3 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 → S 1 h:S^1\to S^1 is a topological conjugacy between f f and g g and h h maps the critical points of f f to the critical points of g g , then h 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 preprint, 2014. However, unlike their proof, which relies on heavy complex-analytic machinery, our proof uses purely real-variable methods and is valid for non-integer critical exponents as well. We do not require h h to preserve the power-law exponents at corresponding critical points.

Some remarks on the optimality of the Bruno-Rüssmann condition

We prove that the Bruno-Russmann condition is optimal for the analytic preservation of a quasi-periodic invariant curve for an analytic twist map. The proof is based on Yoccoz's corresponding result for analytic circle diffeomorphisms and the uniqueness of invariant curves with a given irrational rotation number. We also prove a similar result for analytic Tonelli Hamiltonian flow with n = 2 degrees of freedom; for n ≥ 3 we only obtain a weaker result which recovers and slightly improves a theorem of Bessi.

Descriptive set theory for expansive systems

Kato [5] and Artigue [3] merged the theory of expansive systems [10] and foliations with the continuum theory [14]. Here we merge the expansive systems but with the descriptive set theory [6] instead. More precisely, we define meagre-expansivity for both homeomorphisms and measures by requiring the interior of the dynamical balls up to some prefixed radio to be either empty or with zero measure respectively. We first prove that every cw-expansive homeomorphism of a locally connected metric space without isolated points is meagre-expansive (but not conversely). Second that a homeomorphism of a metric space is meagre-expansive if and only if every Borel probability measure is meagre-expansive. Next that the space of meagre-expansive measures of a homeomorphism of a compact metric space X is an Fσ subset of the space of Borel probability measures equipped with the weak* topology. In the sequel we prove that every homeomorphism with a meagre-expansive measure of a compact metric space has an invariant meagre-expansive measure. Also that the set of periodic points of every meagre-expansive homeomorphism of a compact metric space has empty interior. In the circle or the interval we prove that there are no meagre-expansive homeomorphisms of the circle or the interval. Moreover, the meagre-expansive measures of an interval homeomorphism or a circle homeomorphism with rational rotation number are precisely the finite convex combinations of Dirac measures supported on isolated periodic points. A circle homeomorphism with irrational rotation number has a meagre-expansive measure if and only if it is a Denjoy map. In such a case the meagre-expansive measures are precisely those measures supported on the unique minimal set of the map. To obtain some of our results we will consider a measurable version of the classical Baire Category.

Non-rigidity for circle homeomorphisms with several break points

In this work, we consider two class $P$-homeomorphisms, $f$ and $g$, of the circle with break point singularities, that are differentiable maps except at some singular points where the derivative has a jump. Assume that they have the same irrational rotation number of bounded type and that the derivatives $\text{Df}$ and $\text{Dg}$ are absolutely continuous on every continuity interval of $\text{Df}$ and $\text{Dg}$, respectively. We show that if $f$ and $g$ are not break-equivalent, then any topological conjugating $h$ between $f$ and $g$ is a singular function, i.e., it is continuous on the circle, but $\text{Dh}(x)=0$ almost everywhere (a.e.) with respect to the Lebesgue measure. In particular, this result holds under some combinatorial assumptions on the jumps at break points. It also generalizes previous results obtained for one and two break points and complements that of Cunha–Smania which was established for break equivalence.

Kinematic expansive suspensions of irrational rotations on the circle

We shall show that the rotation of some irrational rotation number on the circle admits suspensions which are kinematic expansive.