Jacek Graczyk, Grzegorz Swiatek
Critical circle homeomorphisms have an invariant measure totally singular with respect to the Lebesgue measure. We prove that singularities of the invariant measure are of Holder type. The Hausdorff dimension of the invariant measure is less than 1 but greater than 0.
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We study the renormalization operator of circle homeomorphisms with a break point and show that it possesses a hyperbolic horseshoe attractor.
We give some results about the dynamics of a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. Our main results concern (1) singularities and (2) the dynamics in the plane that contains the circle. The study presented here is purely analytic.
As an application of the Gordon lemma for orthogonal polynomials on the unit circle, we prove that for a generic set of quasiperiodic Verblunsky coefficients the corresponding two-sided CMV operator has purely singular continuous spectrum. We use a similar argument to that of the Boshernitzan-Damanik result that establishes the corresponding theorem for the discrete Schrödinger operator.
Let $T_{f}$ be a circle homeomorphism with two break points $a_{b},c_{b}$ and irrational rotation number $\varrho_{f}$. Suppose that the derivative $Df$ of its lift $f$ is absolutely continuous on every connected interval of the set $S^{1}\backslash\{a_{b},c_{b}\}$, that $DlogDf \in L^{1}$ and the product of the jump ratios of $ Df $ at the break points is nontrivial, i.e. $\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\frac{Df_{-}(c_{b})}{Df_{+}(c_{b})}\neq1$. We prove that the unique $T_{f}$- invariant probability measure $μ_{f}$ is then singular with respect to Lebesgue measure $l$ on $S^{1}$.
Let $f$ and $g$ be two class $P$-homeomorphisms of the circle $S^{1}$ with break points singularities. Assume that the derivatives $\textrm{Df}$ and $\textrm{Dg}$ are absolutely continuous on every continuity interval of $\textrm{Df}$ and $\textrm{Dg}$ respectively. Denote by $C(f)$ the set of break points of $f$. For $c\in S^{1}$, denote by $π_{s, O_{f}(c)}(f)$ the product of $f-$ jumps in break points lying to the $f-$ orbit of $c$ and by $\textrm{SO}(f) = \{O_{f}(c):~c \in C(f)~\textrm{and}~π_{s, O_{f}(c)}(f)\neq 1\}$, called the set of singular $f$-orbits. The maps $f$ and $g$ are called b…
Stationary measures on the circle from hyperbolic surfaces with cusps cannot be straightened by quasi-symmetries
本文证明：由带尖点的有限面积双曲曲面基本群上随机游走产生的、满足尖点游走指数衰减的圆周平稳测度，其关于勒贝格测度的奇异性在拟对称映射下保持稳定，从而解决了McMullen猜想。
It is well-known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which is purely singular with respect to Lebesgue measure. Can such a map leave invariant an infinite, $σ$-finite invariant measure which is absolutely continuous with respect to Lebesgue measure? In this paper, using an old criterion due to Katznelson, we show that the answer to this question is no.
The rigidity theory for circle homeomophisms with breaks was studied intensively in the last 20 years. It was proved that under mild conditions of the Diophantine type on the rotation number any two $C^{2+α}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-ν}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from th…
For a certain parametrized family of maps on the circle with critical points and logarithmic singularities where derivatives blow up to infinity, we construct a positive measure set of parameters corresponding to maps which exhibit nonuniformly expanding behavior. This implies the existence of "chaotic" dynamics in dissipative homoclinic tangles in periodically perturbed differential equations.
$C^1$-Diffeomorphism Class of some Circle Maps with a Flat Interval
研究平坦段边界奇点次数不同的圆周映射类，证明其拓扑共轭若为双Lipschitz同胚则必为$C^1$微分同胚，从而两类等价。
We study invariant ergodic measures for quasiperiodically forced circle homeomorphisms and derive that either the system is uniquely ergodic or any such measure is associated to some invariant multigraph.
In this paper we show that generic continuous Lebesgue measure preserving circle maps have the s-limit shadowing property. In addition we obtain that s-limit shadowing is a generic property also for continuous circle maps. In particular, this implies that classical shadowing, periodic shadowing and limit shadowing are generic in these two settings as well.
Given any Liouville number $α$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $α$.
We express continuous $\times p,\times q$-invariant measures on the unit circle via some simple forms. On one hand, a continuous $\times p,\times q$-invariant measure is the weak-$*$ limit of average of Dirac measures along an irrational orbit. On the other hand, a continuous $\times p,\times q$-invariant measure is a continuous function on $[0,1]$ satisfying certain function equations.
From pre-lamination to foliated plane
本文研究了平面奇异叶面与圆圈预层状结构之间的对应关系，确定了哪些预层状结构是叶面的无穷远边界，并构建了对应的唯一叶面。
We consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number $τ$ there exists a $C^\infty$ circle diffeomorphism with rotation number $τ$ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value $0\le β\le 1$.
We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsedà and Misiurewicz and contributing to a question of Navas on the absolute continuity of stationary measures, considered in the setup of semigroups of piecewise affine circle homeomorphisms. We focus on the case of resonant boundary derivatives.
In this paper, we prove that for real analytic expanding circle maps, all optimizing measures of a real analytic potential function have zero entropy, unless the potential is cohomologous to constant. We use the group structure of the symbolic space to solve a transversality problem involved. We also discuss applications to optimizing measures for generic smooth potentials and to Lyapunov optimizing measures.
On the Hausdorff dimension of invariant measures for multicritical circle maps
本文为无周期点的$C^3$多临界圆映射的唯一不变测度提供了仅依赖旋转数算术性质的Hausdorff维数显式界。
Topology of the space of measure-preserving transformations of the circle
本文证明了保持Lebesgue测度的2度圆周扩张映射空间同胚于基本群为$\mathbb{Z}$的无穷维Lie群，并提出了一般猜想。
We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic piecewise affine circle homeomorphism with two break points -with respect to the Lebesgue measure- is Morse-Smale.
Automorphic measures and invariant distributions for circle dynamics
本文证明了多临界圆周映射在任意指数下存在唯一的自守测度，并据此证明了其1阶不变分布空间为一维，同时改进了相关的 Denjoy-Koksma 不等式。
On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms
本文证明具有无理旋转数和零平均非线性的充分正则分段光滑圆同胚的唯一不变测度通常具有零Hausdorff维数，并借助广义区间交换变换的组合旋转数概念编码该一般性条件。
Authomorphic measures with negative exponents for multicritical circle maps
本文将多临界圆映射自守$s$-测度的存在唯一性结果从$s>0$拓展至负指数情形，并应用该结论证明了多临界映射族中无理Arnold舌的光滑性结果。
We characterize the maximal entropy measures of partially hyperbolic C^2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons. In the special case of 3-dimensional nilmanifolds other than the torus, this entails the following dichotomy: either the diffeomorphism is a rotation extension of an Anosov diffeomorphism -- in which case there is a unique maximal measure, with full support and zero center Lyapunov exponents -- or there exist exactly two ergodic maximal measures, both hyperbolic and whose center Lyapunov …
Geometric models and Teichmüller 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 Techmüller metric. Moreover, the spaces of geometric models as well as the Teichmüller spaces can be described as the space of Hölder 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 G…
Dimension theory of group actions by circle diffeomorphisms I: Stationary measures
该论文研究圆周光滑随机游走平稳测度的维数性质，为同主题第二部分提供技术工具，还得到相关逼近定理与圆周作用时间反转对称性结论。
On the existence of invariant absolutely continuous probability measures for $C^1$ expanding maps of the circle
本文针对圆周 $C^1$ 一致扩张映射，构造了具有指定最优连续模及绝对连续不变概率测度或保持 Lebesgue 测度的不可数多例，并证明其中许多映射具有无界畸变。
Smooth Circle Covering with a Physical Measure on a Hyperbolic Repelling Fixed Point
本文构造了一个光滑圆覆盖映射的例子，该映射拓扑上与二倍映射等价，并在超几何排斥固定点上有物理测度。通过放松单一一点的光滑条件，还构造了一个物理测度的吸引域具有全测度的例子。
It is well known that real measures on the circle are characterized by their Herglotz transform, an analytic function in the unit disc. Invariance of the measure under N-multiplication translates into a functional equation for the Herglotz transform. Using elements from the theory of Hardy spaces one gets a somewhat surprising condition for a sequence of complex numbers to be the Fourier coefficients of an N-invariant measure. Next, starting from any atomless measure on the circle we construct atomless premesures of bounded kappa-variation in the sense of Korenblum which are invariant under s …
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