Circle Homeomorphisms Research Articles

Anderson localization for Schrödinger operators with monotone potentials over circle homeomorphisms

In this paper, we prove pure point spectrum for a large class of Schrödinger operators over circle maps with conditions on the rotation number going beyond the Diophantine. More specifically, we develop the scheme to obtain pure point spectrum for Schrödinger operators with monotone bi-Lipschitz potentials over orientation-preserving circle homeomorphisms with Diophantine or weakly Liouville rotation number. The localization is uniform when the coupling constant is large enough.

Homeomorphisms of the real line with singularities.

Given a real number a ≠ 0, we consider the set of homeomorphisms f: R\{0}→ R \{a} where{(x, y):x=0}is a vertical asymtote, {(x, y):y=a} is a horizontal asymtote and f is strictly increasing in each connected component (−∞,0) and (0,+∞). In this context, similar to circle homeomorphisms, all possible dynamics are shown. It is established the relationship between existence of periodic orbits and the limit sets. Also, whenever f−n(0) ≠a for all n ∈ N, then the non-existence of periodic orbits leads to a non-trivial limit set, which is either the whole line R or perfect and nowhere dense. It is shown a notion of separation of points that leads to transitivity

Circle Homeomorphisms with Square Summable Diamond Shears

Abstract We introduce and study the space of homeomorphisms of the circle (up to Möbius transformations), which are in $\ell ^{2}$ with respect to modular coordinates called diamond shears along the edges of the Farey tessellation. Diamond shears are related combinatorially to shear coordinates and are also closely related to the $\log \Lambda $-lengths of decorated Teichmüller space introduced by Penner. We obtain sharp results comparing this new class to the Weil–Petersson class and Hölder classes of circle homeomorphisms. We also express the Weil–Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.

Douady–Earle Extensions of Circle Homeomorphisms with One-Point Differentiability at a Hölder Convergence Rate

Douady–Earle Extensions of Circle Homeomorphisms with One-Point Differentiability at a Hölder Convergence Rate

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.

Characterizations of circle homeomorphisms of different regularities in the universal Teichmüller space

In this survey, we first give a summary of characterizations of circle homeomorphisms of different regularities (quasisymmetric, symmetric, or C^{1+\alpha} ) in terms of Beurling–Ahlfors extension, Douady–Earle extension, and Thurston's earthquake representation of an orientation-preserving circle homeomorphism. Then we provide a brief account of characterizations of the elements of the tangent spaces of these sub-Teichmüller spaces at the base point in the universal Teichmüuller space. We also investigate the regularity of the Beurling–Ahlfors extension BA(h) of a C^{1+\mathrm{Zygmund}} orientation-preserving diffeomorphism h of the real line, and show that the Beltrami coefficient \mu (BA(h))(x+iy) vanishes as O(y) uniformly on x near the boundary of the upper half plane if and only if h is C^{1+\mathrm{Lipschitz}} . Finally, we show this criterion is indeed true when h is started with any homeomorphism of the real line that is a lifting map of an orientation-preserving circle homeomorphism.

Estimating Hausdorff measure for Denjoy maps

By employing the recurrence method worked out in Pawelec (2019 arXiv: 1911.12992), we provide effective lower estimates of the proper–dimensional Hausdorff measure of minimal sets of circle homeomorphisms that are not conjugate to any rotation.

Polynomial Entropy of Induced Maps of Circle and Interval Homeomorphisms

We compute the polynomial entropy of the induced maps on hyperspace for a homeomorphism $f$ of an interval or a circle with finitely many non-wandering points.

Circle homeomorphisms with breaks with no $\boldsymbol{C^{2-\nu}}$ conjugacy

Circle homeomorphisms with breaks with no $\boldsymbol{C^{2-\nu}}$ conjugacy

Luzin’s Problem on Fourier Convergence and Homeomorphisms

We show that for every continuous function $$f$$ there exists an absolutely continuous circle homeomorphism $$\phi$$ such that the Fourier series of $$f\circ\phi$$ converges uniformly. This resolves a problem posed by N. N. Luzin.

Angular Values of Nonautonomous and Random Linear Dynamical Systems: Part I---Fundamentals

We introduce the notion of angular values for deterministic linear difference equations and ran- dom linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than $\frac{\pi}{2}$. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.

Symmetric rigidity for circle endomorphisms having bounded geometry

Let f f and g g be two circle endomorphisms of degree d ≥ 2 d\geq 2 such that each has bounded geometry, preserves the Lebesgue measure, and fixes 1 1 . Let h h fixing 1 1 be the topological conjugacy from f f to g g . That is, h ∘ f = g ∘ h h\circ f=g\circ h . We prove that h h is a symmetric circle homeomorphism if and only if h = I d h=Id .

Distortion in the group of circle homeomorphisms

AbstractLet G be the group $\text {PAff}_{+}(\mathbb R/\mathbb Z)$ of piecewise affine circle homeomorphisms or the group ${\operatorname {\mathrm {Diff}}}^{{\kern1pt}\infty }(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in G is given.

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.

Curlicues generated by circle homeomorphisms

We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as boundedness, superficiality, local discrete radius of curvature are linked with dynamical properties of the circle homeomorphism which generates them: rotation number and its continued fraction expansion, existence of a continuous solution of the corresponding cohomological equation and displacement sequence along the orbit.

ρ-bounded orbits and Arnold tongues for quasiperiodically forced circle maps* *Supported by the National Natural Science Foundation of China (11771316, 11790274 and 12171347).

For quasiperiodically forced (qpf) continuous circle maps with degree one, the set of fibred rotation numbers is a closed interval or a singleton. We show in this paper that if a number ρ is in the interior of this interval, then there exist ρ-bounded orbits. We also discuss extensions of the structure of Arnold tongues for qpf circle homeomorphisms to qpf continuous circle maps with degree one.

The Thermodynamic Formalism and the Central Limit Theorem for Stochastic Perturbations of Circle Maps with a Break

Let $T\in C^{2+\varepsilon}(S^{1}\setminus\{x_{b}\})$, $\varepsilon>0$, be an orientation preserving circle homeomorphism with rotation number $\rho_{T}=[k_{1},k_{2},\ldots,k_{m},1,1,\ldots]$, $m\geq 1$, and a single break point $x_{b}$. Stochastic perturbations $\overline{z}_{n+1}=T(\overline{z}_{n})+\sigma\xi_{n+1}$, $\overline{z}_{0}:=z\in S^{1}$ of critical circle maps have been studied some time ago by Diaz-Espinoza and de la Llave, who showed for the resulting sum of random variables a central limit theorem and its rate of convergence. Their approach used the renormalization group technique. We will use here Sinai’s et al. thermodynamic formalism approach, generalised to circle maps with a break point by Dzhalilov et al., to extend the above results to circle homemorphisms with a break point. This and the sequence of dynamical partitions allows us, following earlier work of Vul at al., to establish a symbolic dynamics for any point ${z\in S^{1}}$ and to define a transfer operator whose leading eigenvalue can be used to bound the Lyapunov function. To prove the central limit theorem and its convergence rate we decompose the stochastic sequence via a Taylor expansion in the variables $\xi_{i}$ into the linear term $L_{n}(z_{0})=\xi_{n}+\sum\limits_{k=1}^{n-1}\xi_{k}\prod\limits_{j=k}^{n-1}T^{% \prime}(z_{j})$, ${z_{0}\in S^{1}}$ and a higher order term, which is possible in a neighbourhood $A_{k}^{n}$ of the points $z_{k}$, ${k\leq n-1}$, not containing the break points of $T^{n}$. For this we construct for a certain sequence $\{n_{m}\}$ a series of neighbourhoods $A_{k}^{n_{m}}$ of the points $z_{k}$ which do not contain any break point of the map $T^{q_{n_{m}}}$, $q_{n_{m}}$ the first return times of $T$. The proof of our results follows from the proof of the central limit theorem for the linearized process.

On the Renormalizations of Circle Homeomorphisms with Several Break Points

Let f be an orientation preserving homeomorphism on the circle with several break points, that is, its derivative Df has jump discontinuities at these points. We study Rauzy-Veech renormalizations of piecewise smooth circle homeomorphisms by considering such maps as generalized interval exchange maps of genus one. Suppose that Df is absolutely continuous on each interval of continuity and \(D\ln {Df}\in {\mathbb {L}}_{p}\) for some \(p>1\). We prove that under certain combinatorial assumptions on f, renormalizations \(R^{n}(f)\) are approximated by piecewise Möbius functions in \(C^{1+L_{1}}\)-norm, that means, \(R^{n}(f)\) are approximated in \(C^{1}\)-norm and \(D^{2}R^{n}(f)\) are approximated in \(L_{1}\)-norm. In particular, if the product of the sizes of breaks of f is trivial, then the renormalizations are approximated by piecewise affine interval exchange maps.

Quasisymmetric orbit-flexibility of multicritical circle maps

AbstractTwo given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Świątek, that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$ , then the number of equivalence classes is uncountable (Theorem 1.1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems 1.6 and 1.8).