Orientation-preserving Circle 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.

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.

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.

Kobayashi’s and Teichmüller’s Metrics and Bers Complex Manifold Structure on Circle Diffeomorphisms

Given a modulus of continuity ω, we consider the Teichmuller space TC1+ω as the space of all orientation-preserving circle diffeomorphisms whose derivatives are ω-continuous functions modulo the space of Mobius transformations preserving the unit disk. We study several distortion properties for diffeomorphisms and quasisymmetric homeomorphisms. Using these distortion properties, we give the Bers complex manifold structure on the Teichmuller space TC1+H as the union of TC1+α over all 0 < α ⪯ 1, which turns out to be the largest space in the Teichmuller space of C1 orientation-preserving circle diffeomorphisms on which we can assign such a structure. Furthermore, we prove that with the Bers complex manifold structure on TC1+H, Kobayashi’s metric and Teichmuller’s metric coincide.

Rotation number values at a discontinuity

In the space of orientation-preserving circle maps that are not necessarily surjective nor injective, the rotation number does not vary continuously. Each map where one of these discontinuities occurs is itself discontinuous and we can consider the possible values of the rotation number when we modify this map only at its discontinuities. These values are always rational numbers that necessarily obey a certain arithmetic relation. In this paper we show that in several examples this relation totally characterizes the possible values of the rotation number on its discontinuities, but we also prove that in certain circumstances this relation is not sufficient for this characterization.

Discontinuities of the rotation number

The rotation number of orientation-preserving circle maps that are not necessarily surjective nor injective is discontinuous. In this paper we characterize the circle maps that are points of discontinuity of the rotation number and the relationship between its various possible values on a discontinuity. In particular, we show that, for each circle map corresponding to a discontinuity of the rotation number, all orbits are periodic after a fixed number of iterates, and the entire range of possible rotation numbers at each discontinuity is finite.

Dynamics of incommensurate structures of the\\ Frenkel-Kontorova model

In this paper, we study the dynamics of incommensurate structures of the Frenkel-Kontorova model. When the mean spacing of the particles is irrational, we say that the system has incommensurate structures, for which the phase space of the system is infinite-dimensional. We show that under the overdamped condition, there exists a one-dimensional manifold which is invariant both for the Poincare map and space shifts. Moreover, the Poincare map and space shifts induce orientation-preserving circle homeomorphisms on this manifold with rotation numbers connecting respectively with the average velocity and the mean spacing.

On the Singularity of the Conjugation Between Piecewise-Smooth Circle Homeomorphisms

Let $$f_1$$ and $$f_2$$ be two orientation-preserving circle homeomorphisms with the same irrational rotation number $$\rho $$ and each with a single break point $$b_1$$ and $$b_2$$ , respectively. Suppose that the derivatives $$Df_1$$ and $$Df_2$$ satisfy a certain Zygmund condition except break points and the jumps $$\sigma (b_1)=\frac{Df_1(b_1-0)}{Df_1(b_1+0)}$$ , $$\sigma (b_2)=\frac{Df_2(b_2-0)}{Df_1(b_2+0)}$$ do not coincide. Then the map $$\psi $$ conjugating $$f_1$$ and $$f_2$$ is singular.

Renormalization of circle diffeomorphisms with a break-type singularity**This paper is dedicated to Professors Akhtam Abdurakhmanovich Dzhalilov and Konstantin Mikhailovich Khanin on the occasion of their 60th birthdays.

Let f be an orientation-preserving circle diffeomorphism with an irrational rotation number and with a break point that is, its derivative has a jump discontinuity at this point. Suppose that satisfies a certain Zygmund condition dependent on a parameter We prove that the renormalizations of f are approximated by Möbius transformations in C1-topology if and in C2-topology if Moreover, it is shown that, in case of the coefficients of Möbius transformations get asymptotically linearly dependent. Further, consider two circle diffeomorphisms with a break point, with the same size of the break and satisfying Zygmund condition with We prove that, under a certain technical condition on rotation numbers, the renormalizations of these diffeomorphisms approach each other in C2-topology.

On some generalizations of skew-shifts on

In this paper we investigate maps of the two-torus $\mathbb{T}^{2}$ of the form $T(x,y)=(x+\unicode[STIX]{x1D714},g(x)+f(y))$ for Diophantine $\unicode[STIX]{x1D714}\in \mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\rightarrow \mathbb{T}$, where each $g$ is strictly monotone and of degree 2 and each $f$ is an orientation-preserving circle homeomorphism. For our class of $f$ and $g$, we show that $T$ is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two $T$-invariant graphs. One of the graphs is a strange non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in $\mathbb{T}^{2}$. Only a low-regularity assumption (Lipschitz) is needed on the maps $f$ and $g$, and the results are robust with respect to Lipschitz-small perturbations of $f$ and $g$.

Oscillating sequences, MMA and MMLS flows and Sarnak’s conjecture

We define anoscillating sequence, an important example of which is generated by the Möbius function in number theory. We also define aminimally mean attractable (MMA) flowand aminimally mean-L-stable (MMLS) flow. One of the main results is that any oscillating sequence is linearly disjoint from all MMA and MMLS flows. In particular, this confirms Sarnak’s conjecture for all MMA and MMLS flows. We provide several examples of flows that are MMA and MMLS. These examples include flows defined by all$p$-adic polynomials of integral coefficients, all$p$-adic rational maps with good reduction, all automorphisms of the$2$-torus with zero topological entropy, all diagonalizable affine maps of the$2$-torus with zero topological entropy, all Feigenbaum maps, and all orientation-preserving circle homeomorphisms.

On Möbius orthogonality for interval maps of zero entropy and orientation-preserving circle homeomorphisms

We will prove Sarnak’s conjecture on Mobius disjointness for continuous interval maps of zero entropy and also for orientation-preserving circle homeomorphisms by reducing these result to a well-known theorem of Davenport from 1937.

Cross-ratio distortion and Douady-Earle extension: II. Quasiconformality and asymptotic conformality are local

Let f be an orientation-preserving circle homeomorphism and Φ the Douady-Earle extension of f. In this paper, we show that the quasiconformality and asymptotic conformality of Φ are local properties; i.e., if f is quasisymmetric or symmetric on an arc of the unit circle, then Φ is quasiconformal or asymptotically conformal nearby. Furthermore, our methods enable us to conclude the global quasiconformality and asymptotic conformality from local properties. In the quasiconformal case, our methods also enable us to provide an upper bound for the maximal dilatation of Φ on a neighborhood of the arc in the open unit disk in terms of the cross-ratio distortion norm of f on the arc.

Cross-ratio distortion and Douady-Earle extension: I. A new upper bound on quasiconformality

Let f be an orientation-preserving circle homeomorphism and Φ (f) the Douady–Earle extension of f, and let ||f||cr be the cross-ratio distortion norm of f and K(Φ (f)) be the maximal dilatation of Φ (f). As a consequence of results in [A. Douady and C. J. Earle, ‘Conformally natural extension of homeomorphisms of circle’, Acta Math. 157 (1986) 23–48 and M. Lehtinen, ‘The dilatation of Beurling–Ahlfors extensions of quasisymmetric functions’, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983) 187–191], ln K(Φ (f)) has an upper bound depending on ||f||cr exponentially. In this paper, we first show that ln K(Φ (f)) has an upper bound depending on ||f||cr linearly. Then we extensively study the Douady–Earle extension of a very simple map fλ which depends on a real non-negative parameter λ. For this example, we show that, as λ→∞, (1) our new upper bound on ln K(Φ (fλ)) is substantially smaller than the one given by Douady and Earle in terms of the maximal dilatation of the extremal extension of fλ, and (2) the Douady–Earle extension Φ (fλ) stays exponentially far away from being extremal. Finally, we show that, in general, our upper bound on ln K(Φ (f)) implies the one of Douady and Earle when ||f||cr is large enough.

Herman’s theory revisited

We prove that a $C^{2+\alpha}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\delta$, $0<\delta<\alpha\le1$, is $C^{1+\alpha-\delta}$-smoothly conjugate to a rigid rotation. We also derive the most precise version of Denjoy's inequality for such diffeomorphisms.

On a question of Katok in one-dimensional case

We prove that a $C^{1}$ orientation-preserving circle endomorphism which is Hölder conjugate to a $C^{1}$ circle expanding endomorphism is itself expanding.

Rotation numbers for quasiperiodically forced circle maps-mode-locking vs. strict monotonicity

We describe the relation between the dynamical properties of a quasiperiodically forced orientation-preserving circle homeomorphism f f and the behaviour of the fibred rotation number with respect to strictly monotone perturbations. Despite the fact that the dynamics in the forced case can be considerably more complicated, the result we obtain is in perfect analogy with the one-dimensional situation. In particular, the fibred rotation number behaves strictly monotonically whenever the rotation vector of f f is irrational, which answers a question posed by Herman (1983). In addition, we obtain the continuous structure of the Arnold tongues in parameter families such as the quasiperiodically forced Arnold circle map.

Rotation Numbers of Discontinuous Orientation-Preserving Circle Maps

We extend a few well-known results about orientation preserving homeomorphisms of the circle to orientation preserving circle maps, allowing even an infinite number of discontinuities. We define a set-valued map associated to the lift by filling the gaps in the graph, that shares many properties with continuous functions. Using elementary set-valued analysis, we prove existence and uniqueness of the rotation number, periodic limit orbit in the case when the latter is rational, and Cantor structure of the unique limit set when the rotation number is irrational. Moreover, the rotation number is found to be continuous with respect to the set-valued extension if we endow the space of such maps with the Haussdorff topology on the graph. For increasing continuous families of such maps, the set of parameter values where the rotation number is irrational is a Cantor set (up to a countable number of points).

Scaling exponents, relations, and order dependence for circle maps

Scaling relations describing devil's staircase structures obtained by iterations of orientation-preserving circle maps with minimal slope converging to zero have been investigated. In the limit, the circle map has a zero slope inflection point and the influence of the order at this point on the scaling exponents is analyzed. We study the behavior of the step sizes and the minimal distance to an integer from a (non-zero) cycle point for the super-stable orbit, and in particular we treat the average behavior. This leads to a definition of an ‘average crossover scale exponent’ ν and an ‘average measure exponent’ β , besides the earlier defined ‘average dimension’ D . We find ν ∼- ν, β ∼- β, and D ∼- D. Moreover, β = (1 − D) ν . Finally, as the order increases we observe that hourglass structures evolve in the Arnold tongues.