Attracting current and equilibrium measure for attractors on P^k

On the size of attractors in $\mathbb{CP}^k$

Let $f$ be a holomorphic endomorphism of $\mathbb{CP}^k$ having an attracting set $A$. In this paper, we address the question of the "size" of $A$ in a pluripolar sense. We introduce a conceptually simple framework to have non-algebraic attracting sets. We prove that adding a dimensional condition, these sets support a closed positive current with bounded quasi-potential (which answers a question from T.C. Dinh). Therefore, they are not pluripolar. Moreover, the examples are abundant on $\mathbb{CP}^k$.

Elliptic Curves as Attractors in ${\mathbb P}^2$ Part 1: Dynamics

A study of rational maps of the real or complex projective plane of degree two or more, concentrating on those which map an elliptic curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of exotic dynamical behaviors which are perhaps familiar to the applied dynamics community but not to specialists in several complex variables. For example, we describe smooth attractors with riddled or intermingled attracting basins, and we observe ``blowout'' bifurcations when the transverse Lyapunov exponent for the invariant curve changes sign. In th…

Lattice Structures for Attractors II

The algebraic structure of the attractors in a dynamical system determine much of its global dynamics. The collection of all attractors has a natural lattice structure, and this structure can be detected through attracting neighborhoods, which can in principle be computed. Indeed, there has been much recent work on developing and implementing general computational algorithms for global dynamics, which are capable of computing attracting neighborhoods efficiently. Here we address the question of whether all of the algebraic structure of attractors can be captured by these methods.

Rotation intervals and entropy on attracting annular continua

We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$-robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is uniformly large but the entropy approaches zero as much as desired. The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.

Attractors and Attracting Neighborhoods for Multiflows

We already know a great deal about dynamical systems with uniqueness in forward time. Indeed, flows, semiflows, and maps (both invertible and not) have been studied at length. A view that has proven particularly fruitful is topological: consider invariant sets (attractors, repellers, periodic orbits, etc.) as topological objects, and the connecting sets between them form gradient like flows. In the case of systems with uniqueness in forward time, an attractor in one system is related to nearby attractors in a family of other, "close enough" systems. One way of seeing that connection is through…

On the set of wild points of attracting surfaces in $\mathbb{R}^3$

Suppose that a closed surface $S \subseteq \mathbb{R}^3$ is an attractor, not necessarily global, for a discrete dynamical system. Assuming that its set of wild points $W$ is totally disconnected, we prove that (up to an ambient homeomorphism) it has to be contained in a straight line. Using this result and a modification of the classical construction of a wild sphere due to Antoine we show that there exist uncountably many different $2$--spheres in $\mathbb{R}^3$ none of which can be realized as an attractor for a homeomorphism.

Power law asymptotics in the creation of strange attractors in the quasi-periodically forced quadratic family

Let $Φ$ be a quasi-periodically forced quadratic map, where the rotation constant $ω$ is a Diophantine irrational. A strange non-chaotic attractor (SNA) is an invariant (under $Φ$) attracting graph of a nowhere continuous measurable function $ψ$ from the circle $\mathbb{T}$ to $[0,1]$. This paper investigates how a smooth attractor degenerates into a strange one, as a parameter $β$ approaches a critical value $β_0$, and the asymptotics behind the bifurcation of the attractor from smooth to strange. In our model, the cause of the strange attractor is a so-called torus collision, whereby an attr…

环面吸引子熵的普适下界

Universal bounds on the entropy of toroidal attractors

本文研究环面吸引子的拓扑熵，依据纯拓扑性质给出下界，证明其光滑吸引动力学要么熵至少为 $\log 2$，要么存在平稳动力学使熵为零。

吸引测度

Attracting measures

本文提出“吸引测度”概念，将Axiom A吸引子上SRB测度的Lebesgue绝对连续测度前推弱收敛性质抽象为拓扑框架下的结果，强调该性质与物理测度同等重要。

Partial hyperbolicity and attracting regions in 3-dimensional manifolds

This thesis attempts to contribute to the study of differentiable dynamics both from a semi-local and global point of view. The center of study is differentiable dynamics in manifolds of dimension 3 where we are interested in the understanding of the existence and structure of attractors as well as dynamical and topological implications of the existence of a global partially hyperbolic splitting. The main contributions are new examples of dynamics without attractors where we get a quite complete description of the dynamics around some wild homoclinic classes and two results on dynamical cohere…

一类混合系统的吸引子格计算

Computation of Attractor Lattices for a Class of Hybrid Systems

该研究针对紧致度量空间上的一类混合系统，证明其吸引子格同构于关联半流的吸引子格，通过网格构造实现混合吸引子子格的外近似，还计算了多个系统的Conley-Morse图与吸引子格。

Surface Attractors

Let $f$ be a continuous endomorphism of a surface $M$, and $A$ an attracting set such that the restriction $f|_A: A \to A$ is a $d:1$ covering map. We show that if $f$ is a local homeomorphism in the immediate basin $B^0_A$ of $A$, then $f$ is also a $d:1$ covering of $B^0_A$.

Torus-breakdown near a Bykov attractor: a case study

There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of vector fields unfolding an attracting heteroclinic network, linking two saddle-foci with $ (\mathbb{SO}(2) \oplus \mathbb{Z}_2)$-symmetry. The vector field is the restriction to $\mathbb{S}^3$ of a polynomial vector field in $\mathbb{R}^4$. We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a phenomenon called Torus-Breakdown theory.…

Speed of convergence towards attracting sets for endomorphisms of P^k

Let f be a non-invertible holomorphic endomorphism of P^k having an attracting set A. We show that, under some natural assumptions, A supports a unique invariant positive closed current τ, of the right bidegree and of mass 1. Moreover, if R is a current supported in a small neighborhood of A then its push-forwards by f^n converge to τexponentially fast. We also prove that the equilibrium measure on A is hyperbolic.

Equilibrium states for certain partially hyperbolic attractors

We prove that a class of partially hyperbolic attractors introduced by Castro and Nascimento have unique equilibrium states for natural classes of potentials. We also show if the attractors are $C^2$ and have invariant stable and centerunstable foliations, then there is a unique equilibrium state for the geometric potential and its 1-parameter family. We do this by applying general techniques developed by Climenhaga and Thompson.

Uniqueness of equilibrium states for Lorenz attractors in any dimension

In this note, we consider the thermodynamic formalism for Lorenz attractors of flows in any dimension. Under a mild condition on the Hölder continuous potential function $ϕ$, we prove that for an open and dense subset of $C^1$ vector fields, every Lorenz attractor supports a unique equilibrium state. In particular, we obtain the uniqueness for the measure of maximal entropy.

Robust Exponential Mixing and Convergence to Equilibrium for Singular Hyperbolic Attracting Sets

We extend results on robust exponential mixing for geometric Lorenz attractors, with a dense orbit and a unique singularity, to singular-hyperbolic attracting sets with any number of (either Lorenz- or non-Lorenz-like) singularities and finitely many ergodic physical/SRB invariant probability measures, whose basins cover a full Lebesgue measure subset of the trapping region of the attracting set. We obtain exponential mixing for any physical probability measure supported in the trapping region and also exponential convergence to equilibrium, for a $C^2$ open subset of vector fields in any $d$-…

经典 Lorenz 吸引子与高维截面双曲吸引子的平衡态

Equilibrium states for the classical Lorenz attractor and sectional-hyperbolic attractors in higher dimensions

本文证明在包含经典 Lorenz 吸引子的 $C^1$ 开稠向量场族中，截面双曲吸引子上的 Hölder 势函数在奇点质量非平衡态时具有唯一平衡态，从而确立唯一最大熵测度。

大部分截面扩张流的物理测度

Physical measures for mostly sectional expanding flows

针对C2向量场的部分双曲吸引集，证明其支撑遍历物理/SRB测度的充要条件为捕获区域在正勒贝格测度子集上存在非一致截面扩张，最多支撑有限个此类测度且为中心不稳定方向吉布斯态，推广微分同胚结果至连续时间系统并给出应用实例。

The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations

An attractor of a piecewise-smooth continuous system of differential equations can bifurcate from a stable equilibrium to a more complicated invariant set when it collides with a switching manifold under parameter variation. Here numerical evidence is provided to show that this invariant set can be chaotic. The transition occurs locally (in a neighbourhood of a point) and instantaneously (for a single critical parameter value). This phenomenon is illustrated for the normal form of a boundary equilibrium bifurcation in three dimensions using parameter values adapted from of a piecewise-linear m…

The well-posedness and the regularity of global attractor for a couple stress fluid through porous layer with the local thermal non-equilibrium effect

In the article, we aim to investigate the well-posedness of solution and the regularity of the global attractor for the couple stress fluid in saturated porous media with the local thermal non-equilibrium effect. To be more specific, we firstly show the existence and uniqueness of global weak solution to the model by making use of the standard Galerkin method. Second, relying on verifying the uniformly compact condition required, we prove the existence of the global attractor of the model in the space where the weak solution resides. Finally, we improve the regularity of global attractor by un…

通过密度构造某些部分超双曲吸引子的平衡态

Constructing equilibrium states for some partially hyperbolic attractors via densities

本文提出一种新方法构造部分双曲系统平衡态，扩展了均匀双曲情形下的吉布斯测度构造，引入新的问题需仔细处理，包括中心稳定流形的有界扩张条件或中心不稳定流形的亚指数收缩条件。

Fourier decay of equilibrium states for bunched attractors

Let $M$ be a closed manifold, and let $f:M \rightarrow M$ be a $C^{2+α}$ Axiom A diffeomorphism. Suppose that $f$ has an attractor $Ω$ with codimension 1 stable lamination. Under a generic nonlinearity condition and a suitable bunching condition, we prove polynomial Fourier decay in the unstable direction for a large class of invariant measures on $Ω$. Our result applies in particular for the measure of maximal entropy. We construct in the appendix an explicit solenoid that satisfies the nonlinearity and bunching assumption.

高余维渐进横向扩张流的物理测度

Physical measures for asymptotically sectional expanding flows in higher co-dimensions

本文研究了高余维渐进横向扩张流中物理测度的存在性，提出了适用于高余维数的吸引子构造方法，并证明了其具有完整遍历盆地的SRB测度。

Uniqueness of the measure of maximal entropy for singular hyperbolic flows in dimension 3 and more results on equilibrium states

We prove that any 3-dimensional singular hyperbolic attractor admits for any Hölder continuous potential $V$ at most one equilibrium state for $V$ among regular measures. We give a condition on $V$ which ensures that no singularity can be an equilibrium state. Thus, for these $V$'s, there exists a unique equilibrium state and it is a regular measure. Applying this for $V\equiv 0$, we show that any 3-dimensional singular hyperbolic attractor admits a unique measure of maximal entropy.

Non-hyperbolic Iterated Function Systems: attractors and stationary measures

We consider iterated function systems $\mathrm{IFS}(T_1,\dots,T_k)$ consisting of continuous self maps of a compact metric space $X$. We introduce the subset $S_{\mathrm{t}}$ of {\emph{weakly hyperbolic sequences}} $ξ=ξ_0\ldotsξ_n \ldots \in Σ_k^+$ having the property that $\bigcap_n T_{ξ_{0}}\circ\cdots\circ T_{ξ_{n}}(X)$ is a point $\{π(ξ)\}$. The target set $π(S_{\mathrm{t}})$ plays a role similar to the semifractal introduced by Lasota-Myjak. Assuming that $S_{\mathrm{t}}\ne \emptyset$ (the only hyperbolic-like condition we assume) we prove that the IFS has at most one strict attractor and…

具有$\mathbb{Z}_4$对称性的系统中三重不稳定性产生新型Lorenz-like吸引子的解析证明

Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with $\mathbb{Z}_4$-symmetry

本文在具有Z4对称性的微分方程系统中证明，平衡态的三重零特征值分岔可产生经典Lorenz吸引子和Simó angel两类伪双曲吸引子，并利用规范形理论为三维Hénon映射中数值观测到的离散Lorenz-like吸引子与Simó angels提供理论证明。

SRB measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus

In this paper we address the existence and ergodicity of non-hyperbolic attracting sets for a certain class of smooth endomorphisms on the solid torus. Such systems allow a formulation as a skew product system defined by planar diffeomorphisms have contraction on average which forced by any expanding circle map. These attractors are invariant graphs of upper semicontinuous maps which support exactly one $SRB$ measure. In our approach, these skew product systems arising from iterated function systems generated by a finitely many weak contractive diffeomorphisms. Under some conditions including …

非吸引子树的计数与排斥点的等分布

Counting non-attracting subtrees and the equidistribution of repelling points

该研究针对非阿基米德完备域上的Berkovich射影直线，计数不含吸引点的非吸引子树，证明其次数和等于有理函数次数，并得到其测度等分布于平衡测度的结果，解答了相关问题。

全局吸引子猜想在特殊情况下的证明

A Proof of the Global Attractor Conjecture in a Special Case

本文在特定结构条件下证明了复平衡质量作用反应网络的全局吸引子猜想，通过两步证明确保正轨迹收敛到唯一正平衡点。