Jaume Llibre, Michael Todd
We characterise the set of fixed points of a class of holomorphic maps on complex manifolds with a prescribed homology. Our main tool is the Lefschetz number and the action of maps on the first homology group.
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On Lefschetz's point-free periodicity
本文提出两种新方法研究Lefschetz无点周期性：一是关注满足Wecken性质的空间（如气泡空间），二是进行相对研究，这对动力系统排斥子等有重要意义。
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.
Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_0$ with Lefschetz index one. Le Roux conjectured that $z_0$ is accumulated by periodic orbits. In this article, we will approach Le Roux's conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_0$ and if the area of $M$ is finite, then $z_0$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_0$ is the limit in weak-star topology of a sequence of invaria…
We show how a variant of the Lefschetz Fixed Point Theorem may be used to count the number of periodic orbits for certain rational difference equations.
On the periodic structure of $C^1$ self-maps on the product of spheres of different dimensions
该研究利用Lefschetz不动点理论，刻画了不同维数球面乘积上三类$C^1$自映射的周期结构，并将结果推广至紧连通单连通单Lie群上的同类映射。
We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then periodic points in the boundary of A are dense in this boundary. To prove this in the non simply- connected or parabolic situations we prove a more abstract, geometric coding trees version.
The generalized Lefschetz number and loop braid groups
本文通过引入环辫群和广义勒夫谢茨数，研究三维流形中同胚的固定点和周期点的拓扑动力学性质。
We consider the homotopical dynamics on compact orientable surfaces of positive genus g. We establish a sufficient and necessary algebraic criterion for homotopy classes with infinitely many periodic points of maps on such surfaces in terms of the characteristic polynomial of the matrix representing the correspondig homomorphism of the first homology group. Our methods differ from that of the literature, which typically uses Nielsen number theory, because Nielsen numbers of iterates are difficult to compute except in special cases where they can be easily defined in terms of the easily computa…
Let $f:(\mathbb{C}^n,0)\mapsto(\mathbb{C}^n,0)$ be a germ of an $n$-dimensional holomorphic map. Assume that the origin is an isolated fixed point of each iterate of $f$. Then $\{\mathcal{N}_q(f)\}_{q=1}^{\infty}$, the sequence of the maximal number of periodic orbits of period $q$ that can be born from the fixed point zero under a small perturbation of $f$, is well defined. According to Shub-Sullivan, Chow-Mallet-Paret-Yorke and G. Y. Zhang, the linear part of the holomorphic germ $f$ determines some natural restrictions on the sequence(cf. Theorem 1.1). Later, I. Gorbovickis proves that when…
Accessible hyperbolic components in anti-holomorphic dynamics
研究反全纯二次族的连通性领域（tricorn）的双曲分量可访问性，发现尽管某些分量不可达，但tricorn中存在无限多个可访问的双曲分量。
Holomorphic motions of weighted periodic points
本文研究复射影空间自同态稳定族中排斥周期点的全纯运动，并证明相关图像在自然测度下的渐近等分布。
This note presents a method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will prove Needham's theorem in a more general version.
Homological data on the periodic structure of self-maps on wedge sums
针对拓扑流形楔和上的连续自映射，本文计算 Lefschetz 数、Dold 系数及代数周期，并揭示环面楔和情形中的同调障碍。
For a class of one-dimensional holomorphic maps f of the Riemann sphere we prove that for a wide class of potentials h the topological pressure is entirely determined by the values of h on the repelling periodic points of f. This is a version of a classical result of Bowen for hyperbolic diffeomorphisms in the holomorphic non-uniformly hyperbolic setting.
We prove that if a positive closed current is bounded by another one with bounded, continuous or Hoelder continuous super-potentials, then it inherits the same property. There are two different methods to define wedge-products of positive closed currents of arbitrary bi-degree on compact Kaehler manifolds using super-potentials and densities. When the first method applies, we show that the second method also applies and gives the same result. As an application, we obtain a sharp upper bound for the number of isolated periodic points of holomorphic maps on compact Kaehler manifolds whose action…
We show that all GL(2, R)-equivariant point markings over orbit closures of primitive genus two translation surfaces arise from marking pairs of points exchanged by the hyperelliptic involution, Weierstrass points, or the golden points in the golden eigenform locus. As corollaries, we classify the holomorphically varying families of points over orbifold covers of genus two Hilbert modular surfaces, solve the finite blocking problem on genus two translation surfaces, and show that there is at most one nonarithmetic rank two orbit closure in the minimal stratum in genus four.
We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if $A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $μ$ is an arbitrary $f$-invariant measure with positive Lyapunov exponents on the boundary of $A$, then $μ$-almost every point $q$ in the boundary of $A$ is accessible along a curve from $A$. In fact we prove the accessability of every "good" $q$ i.e. such $q$ for which "small neighbourhoods arrive at large scale" under iteration of $f$. This generalizes Douady-Eremenko-Levin-Petersen theorem …
Bifurcation of limit cycles in piecewise holomorphic systems separated by two external circles
本文研究由两个外圆分隔的三区域分片全纯系统扰动线性中心时极限环的分支数量，并证明在特定构型下四次系统至少存在十六个极限环。
We show that if f_t is a holomorphic family of quadratic-like maps with all periodic orbits repelling so that for each real t the map f_t is a real Collet-Eckmann S-unimodal map then, writing m_t for the unique absolutely continuous invariant probability measure of f_t, the map t -> \int g dm_t is real analytic for any real analytic function g.
Let $Δ^{2}$ be a ball in the complex vector space $\mathbb{C}^{2}$ centered at the origin, let $f:Δ^{2}\to \mathbb{C}^{2}$ be a holomorphic mapping$,$ with $f(0)=0$, and let $M$ be a positive integer. If the origin 0 is an isolated fixed point of the $M$ th iteration $f^{M}$ of $f,$ then one can define the number $\mathcal{O}_{M}(f,0)$ of periodic orbits of $f$ with period $M$ hidden at the fixed point 0, which has the meaning: any holomorphic mapping $g:Δ^{2}\to \mathbb{C}^{2}$ sufficiently close to $f$ in a neighborhood of the origin has exactly $% \mathcal{O}_{M}(f,0)$ distinct periodic orb…
Let $Δ^{n}$ be the ball $|x|<1$ in the complex vector space $\mathbb{C}% ^{n}$, let $f:Δ^{n}\to \mathbb{C}^{n}$ be a holomorphic mapping and let $M$ be a positive integer. Assume that the origin $% 0=(0,..., 0)$ is an isolated fixed point of both $f$ and the $M$-th iteration $f^{M}$ of $f$. Then for each factor $m$ of $M,$ the origin is again an isolated fixed point of $f^{m}$ and the fixed point index $μ_{f^{m}}(0)$ of $f^{m}$ at the origin is well defined, and so is the (local) Dold's index (see [\ref{Do}]) at the origin:% \begin{equation*} P_{M}(f,0)=\sum_{τ\subset P(M)}(-1)^{#τ}μ_{f^{M:τ}}…
Equidistribution of saddle periodic points for automorphisms on compact Kähler manifolds
本文证明了紧致Kähler流形上具有简单上同调作用的全纯自同构的鞍型周期点关于平衡测度等分布，并推广到正拓扑熵的Kähler曲面情形。
Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms
本文研究全纯1-形式层绝对周期叶状结构的遍历性，证明在温和假设下其遍历且非稠密叶位于可数子轨形并集，并给出等周期形式空间的连通性与覆盖构造例子。
We prove that every finite energy $J$-holomorphic map $u(s,t):\mathbb R\times S^1 \rightarrow {\mathbb R} \times \widetilde{M}$ exponentially converges to a periodic orbit of Reeb vector field of $\widetilde M,$ as $s\to \infty.$
We establish an approximation of the activity current $T_c$ in the parameter space of a holomorphic family $f$ of rational functions having a marked critical point $c$ by parameters for which $c$ is periodic under $f$, i.e., is a superattracting periodic point. This partly generalizes a Dujardin--Favre theorem for rational functions having preperiodic points, and refines a Bassanelli--Berteloot theorem on a similar approximation of the bifurcation current $T_f$ of the holomorphic family $f$. The proof is based on a dynamical counterpart of this approximation.
We show the existence of a dense orbit for real Rel flows on the area-1 locus of every connected component of every stratum of holomorphic 1-forms with at least 2 distinct zeros. For this purpose, we establish a general density criterion for ${\rm SL}(2,\mathbb{R})$-orbit closures, based on finding an orbit of a real Rel flow whose closure contains a horocycle. This criterion can be verified using explicit constructions of holomorphic 1-forms with a periodic horizontal foliation. Our constructions also provide explicit examples of dense leaves of the absolute period foliation and many subfolia…
Let $f:M\rightarrow M$ be a biholomorphisms on two--dimensional a complex manifold, and let $X\subseteq M$ be a compact $f$--invariant set such that $f|X$ is asymptotically dissipative and without sinks periodic points. We introduce a solely dynamical obstruction to dominated splitting, namely critical point. Critical point is a dynamical object and capture many of the dynamical properties of their one--dimensional counterpart.
On the (Dis)connection Between Growth and Primitive Periodic Points
本文将Cornalba-Shiffman关于零阶全纯函数零点快速增长的结果推广至复动力学，证明其孤立原始周期点数量也可任意快速增长，从而回答了Buhovsky等人提出的问题。
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