Circle Endomorphisms Research Articles

Large coupling asymptotics for the Lyapunov exponents of some Schrödinger cocyles over strongly expanding circle endomorphisms

ABSTRACT We quantify the coupling asymptotics for the Lyapunov exponent of the Schrödinger cocycle over strongly expanding maps x ↦ bx ( mod 1 ) on T , for a large class of potential functions. We refine the previous lower bound results for these Lyapunov exponents, to get asymptotic results for all E ∈ R and for sufficiently expanding circle maps.

Monotone Families of Circle Diffeomorphisms Driven by Expanding Circle Maps

We consider monotone families of circle diffeomorphisms forced by the strongly chaotic circle endomorphisms x↦bxmod1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\mapsto bx \\mod 1$$\\end{document}, where the integer b is large. We obtain estimates of the fibered Lyapunov exponents and show that in the limit as b tends to infinity, they approach the values of the Lyapunov exponents for the corresponding random case. The estimates are based on a control of the distribution of the iterates of almost every point, up to a fixed (small) scale, depending on b.

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 .

On the Lyapunov Exponents for a Class of Circle Diffeomorphisms Driven by Expanding Circle Endomorphisms

We consider skew-product maps on mathbb {T}^2 of the form F(x,y)=(bx,x+g(y)) where g:mathbb {T}rightarrow mathbb {T} is an orientation-preserving C^2-diffeomorphism and bge 2 is an integer. We show that the fibred (upper and lower) Lyapunov exponent of almost every point (x, y) is as close to int _mathbb {T}log (g'(eta ))deta as we like, provided that b is large enough.

Geometric Gibbs theory

Geometric models and Teichm\"uller structures have been introduced for the space of smooth expanding circle endomorphisms and for the space of uniformly symmetric circle endomorphisms. The latter one is the completion of the previous one under the Techm\"uller metric. Moreover, the spaces of geometric models as well as the Teichm\"uller spaces can be described as the space of H\"older continuous scaling functions and the space of continuous scaling functions on the dual symbolic space. The characterizations of these scaling functions have been also investigated. The Gibbs measure theory and the dual Gibbs measure theory for smooth expanding circle dynamics have been viewed from the geometric point of view. However, for uniformly symmetric circle dynamics, an appropriate Gibbs measure theory is unavailable, but a dual Gibbs type measure theory has been developed for the uniformly symmetric case. This development extends the dual Gibbs measure theory for the smooth case from the geometric point of view. In this survey article, We give a review of these developments which combines ideas and techniques from dynamical systems, quasiconformal mapping theory, and Teichm\"uller theory. There is a measure-theoretical version which is called $g$-measure theory and which corresponds to the dual geometric Gibbs type measure theory. We briefly review it too.

Positive fibered Lyapunov exponents for some quasi-periodically driven circle endomorphisms with critical points

Positive fibered Lyapunov exponents for some quasi-periodically driven circle endomorphisms with critical points

Positive fibered Lyapunov exponents for some quasi-periodically driven circle endomorphisms with critical points

Positive fibered Lyapunov exponents for some quasi-periodically driven circle endomorphisms with critical points

Simultaneous uniformization for uniformly quasisymmetric circle dynamical systems

In the 1960s Bers showed how to uniformize simultaneously two Riemann surfaces of the same finite analytic type by using a single quasi-Fuchsian group of the first kind. In this paper, we show how to uniformize simultaneously two uniformly quasisymmetric circle endomorphisms of the same degree by a unique normalized branched covering of the Riemann sphere of the same degree such that this branched covering has a unique normalized quasicircle as an invariant limit set. We use this simultaneous uniformization to define a transformation between their spaces of probability invariant measures and formulate several equivalent conjectures to the uniqueness conjecture for symmetric invariant probability measures. In a subsequent paper, we study these conjectures.

Conformally natural extensions in view of dynamics

We give an easy description of the barycentric extension of a map of the unit circle to the closed unit disk using some ideas from dynamical systems. We then prove that every circle endomorphism of the unit circle of degree d ≥ 2 (with a topological expansion condition) has a conformally natural extension to the closed unit disk which is real analytic on the open unit disk. If the endomorphism is uniformly quasisymmetric, then the extension is quasiconformal.

Martingales for quasisymmetric systems and complex manifold structures

We construct an infinite martingale sequence on the dual symbolic space from a uniformly quasisymmetric circle endomorphism preserving the Lebesgue measure. This infinite martingale sequence is uniformly bounded. Thus from the martingale convergence theorem, there is a limiting martingale which is the unique L 1 limit of this uniformly bounded infinite martingale sequence. Moreover, we prove that the classical Hilbert transform gives an almost complex structure on the space of all uniformly quasisymmetric circle endomorphisms preserving the Lebesgue measure. Furthermore, we discuss the complex manifold structure which is the integration of the almost complex structure. We further discuss the comparison between the global Kobayashi's metric and the global Teichmuller metric on the fiber of the forgetful map at the basepoint. We prove that these two metrics are not equivalent.

Strangely dispersed minimal sets in the quasiperiodically forced Arnold circle map

We study quasiperiodically forced circle endomorphisms, homotopic to the identity, and show that under suitable conditions these exhibit uncountably many minimal sets with a complicated structure, to which we refer to as ‘strangely dispersed’. Along the way, we generalize some well-known results about circle endomorphisms to the uniquely ergodically forced case. Namely, all rotation numbers in the rotation interval of a uniquely ergodically forced circle endomorphism are realized on minimal sets, and if the rotation interval has a non-empty interior then the topological entropy is strictly positive. The results apply in particular to the quasiperiodically forced Arnold circle map, which serves as a paradigm example.

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.

Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family

In this paper we revisit once again, see [ShSu], a family of expanding circle endomorphisms. We consider a family {Bθ} of Blaschke products acting on the unit circle, T, in the complex plane obtained by composing a given Blashke product B with the rotations about zero given by mulitplication by θ ∈ T. While the initial map B may have a fixed sink on T there is always an open set of θ for which Bθ is an expanding map. We prove a lower bound for the average measure theoretic entropy of this family of maps in terms of R ln|B′(z)|dz .

Metric Invariants in Dynamical Systems

The scaling function and the linear model for a circle endomorphism are two important smooth invariants under conjugacy. We discuss these two invariants and some relations between them. Furthermore, we use these relations to discuss some realization results in this direction. The discussion in this paper avoids quasiconformal mapping theory and Gibbs theory and g-measure theory, which are used in our previous discussions, therefore, is straightforward and simple.

Critical saddle-node bifurcations and Morse–Smale maps

We study the saddle-node bifurcation in diffeomorphisms with a “critical” homoclinic orbit to the saddle-node point. In a typical family F γ of diffeomorphisms that undergoes a saddle-node bifurcation at γ = 0 , the diffeomorphisms (possibly after reparameterization) for γ < 0 have two periodic points which coalesce for γ = 0 and then disappear for γ > 0 . If the saddle-node points has a critical homoclinic orbit it is known that complicated dynamics can occur for γ > 0 . We show that there are families such that for γ > 0 , there are parameter values arbitrarily close to γ = 0 for which the map is Morse–Smale. Such parameter values are shown to have positive frequency at γ = 0 + . In the process we show that the boundary of the set of Morse–Smale diffeomorphisms possesses comb-like structures. We also show that there are other families unfolding a saddle-node for which there are no Morse–Smale maps for γ > 0 . These results rely heavily on projecting to circle endomorphisms. We conclude with some numerical results from such maps.

Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms

We study the scaling function of a $C^{1+h}$ expanding circle endomorphism. We find necessary and sufficient conditions for a Holder continuous function on the dual symbolic space to be realized as the scaling function of a $C^{1+h}$ expanding circle endomorphism. We further represent the Teichmuller space of $C^{1+h}$ expanding circle endomorphisms by the space of Holder continuous functions on the dual symbolic space satisfying our necessary and sufficient conditions and study the completion of this Teichmuller space in the universal Teichmuller space.

Statistical properties of some spatially discretized dynamical systems

Computer simulations of dynamical systems are {spatial discretizations} in which the finite machine arithmetic space replaces the continuum state space of the original system. Any trajectory of a spatial discretization of a dynamical system is thus eventually periodic, so the dynamical behaviour of such computations are essentially determined by the cycles of the discretized map. Such dynamical behaviour depends seemingly randomly on the fineness of the discretization mesh. In this paper statistical properties of the maximal cycles of spatial discretizations are investigated for some systems such as the tent map, rotations on a circle and toral endomorphisms.