C. Azevedo, H. Cabral, P. Ontaneda
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.
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Dynamics and Rigidity through the Lens of Circles
本文通过研究圆排列中的四个自然问题，探讨了无限体积均匀空间的动力学、几何与刚性的相互作用，揭示了均匀动力学的新前沿。
In this note, we establish an original result for the thermodynamic formalism in the context of expanding circle transformations with an indifferent fixed point. For an observable whose continuity modulus is linked to the dynamics near such a fixed point, by identifying an appropriate linear space to evaluate the action of the transfer operator, we show that there is a strictly positive eigenfunction associated with the maximal eigenvalue given as the exponential of the topological pressure. Taking into account also the corresponding eigenmeasure, the invariant probability thus obtained is pro…
It is well-known that the Manneville-Pomeau map with a parabolic fixed point of the form $x\mapsto x+x^{1+α} \mod 1$ is stochastically stable for $α\ge 1$ and the limiting measure is the Dirac measure at the fixed point. In this paper we show that if $α\in (0,1)$ then it is also stochastically stable. Indeed, the stationary measure of the random map converges strongly to the absolutely continuous invariant measure for the deterministic system as the noise tends to zero.
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。
We consider some smooth maps on a bouquet of circles. For these maps we can compute the number of fixed points, the existence of periodic points and an exact formula for topological entropy. We use Lefschetz fixed point theory and actions of our maps on both the fundamental group and the first homology group.
Gluing polynomials along the circle
本文针对具有同阶有限超吸引不动点的不可重整多项式，证明其可沿超吸引不动点直接吸引盆的若尔当边界，通过粘合构造得到有理映射。
Periodicity and Rotation Number for Random Circle Homeomorphisms
研究随机动力系统中圆同胚的周期性和旋转数，证明随机周期点存在与旋转数有理性的关系，并指出确定性与随机性下的固定点与旋转数整数性的差异。
Amalgamated Free Products of Circle Actions with a Bounded Number of Fixed Points
本文引入圆作用的拼接自由乘积，通过扩张两个作用并重新排列插入区间，证明其在自然轨道和索引条件下定义良好，产生最小圆作用，并在拓扑共轭意义下唯一。同时研究其动力学性质，给出保持固定点数有界及保持莫比乌斯型的条件。
In this paper, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps.
Loewner evolution of hedgehogs and 2-conformal measures of circle maps
研究全纯微分同胚芽在无理不动点处的Siegel紧集族对应的圆映射半群，证明其无穷小生成元由Loewner方程给出，且驱动测度为2-共形测度。
This paper is concerned with the existence of multiple stable fixed point solutions of the homogeneous Kuramoto model. We develop a necessary condition for the existence of stable fixed points for the general network Kuramoto model. This condition is applied to show that for sufficiently dense n-node networks, with node degrees at least 0.9395(n-1), the homogeneous (equal frequencies) model has no non-zero stable fixed point solution over the full space of phase angles in the range -Pi to Pi. This result together with existing research proves a conjecture of Verwoerd and Mason (2007) that for …
We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible to obtain polynomial approximations. Here we develop an algorithm to obtain them as sums of homogeneous functions by solving suitable cohomological equations. We deal with both the differentiable and analytic cases. We also study the dependence on parameters. In the companion paper, Invariant manifolds of parabolic fixed points (I), these approximations are…
We construct a non-Hamiltonian symplectic circle action on a closed, connected, six-dimensional symplectic manifold with exactly 32 fixed points.
Mixing for Free Semigroup Actions of Blaschke Products on the Circle
本文对单位圆盘内带不动点的有限Blaschke乘积及其自由半群作用的混合速率完成分类，揭示混合速率的非刚性及刚性条件，明确双指数混合的尖锐指数值。
The Morse Smale property for time-periodic scalar reaction-diffusion equation on the circle
本文研究圆周上时间周期标量半线性抛物方程的Morse-Smale性质，证明其Poincaré映射双曲不动点无同宿连接、异宿不动点稳定与不稳定流形横截相交，且在所有ω-极限集双曲时方程具备Morse-Smale性质。
We describe a procedure based on the iteration of an initial function by an appropriated operator, acting on continuous functions, in order to get a fixed point. This fixed point will be a calibrated subaction for the doubling map on the circle and a fixed Lipschitz potential. We study analytical and generic properties of this process and we provide some computational evaluations of subactions using a discretization of the circle. The fixed point is unique if the maximizing probability is unique. We proceed a careful analysis of the dynamics of this operator close by the fixed point in order t…
The study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least 1 + dim(M)/2 fixed points; this follows from Morse theory for the momentum map of the action. In this paper we use Atiyah-Bott-Berline-Vergne (ABBV) localization in equivariant cohomology to prove that this conclusion also holds for symplectic circle actions with non-empty fixed sets, as long as the Chern class map is somewhere injective -- the Chern class map assigns to a fixed point the …
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