Andre de Carvalho
We describe a circle of ideas relating the dynamics of 2-dimensional homeomorphisms to that of 1-dimensional endomorphisms. This is used to introduce a new class of maps generalizing that of Thurston's pseudo-Anosov homeomorphisms.
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We show that any pseudo-Anosov map that is a lift of pseudo-Anosov homeomorphism of a nonorientable surface has vanishing SAF invariant. We also provide a criterion to certify that a pseudo-Anosov map is not such a lift.
From quasigeodesic to pseudo-Anosov flows
本文证明了闭合双曲3流形上拟测地流可变形为同时具有拟测地和伪Anosov性质的流。
We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of abelian differentials. The existence of such sequences show that all pseudo-Anosov homeomorphisms fixing orientable measured foliations cannot be obtained by the Rauzy-Veech induction strategy.
We investigate a phenomenon observed by W. Thurston wherein one constructs a pseudo-Anosov homeomorphism on the limit set of a certain lift of a piecewise-linear expanding interval map. We reconcile this construction with a special subclass of generalized pseudo-Anosovs, first defined by de Carvalho. From there we classify the circumstances under which this construction produces a pseudo-Anosov. As an application, we produce for each $g \geq 1$ a pseudo-Anosov $ϕ_g$ on the surface of genus $g$ that preserves an algebraically primitive translation structure and whose dilatation $λ_g$ is a Salem…
The dilatation of a pseudo-Anosov braid is a conjugacy invariant. In this paper, we study the dilatation of a special family of pseudo-Anosov braids. We prove an inductive formula to compute their dilatation, a monotonicity and an asymptotic behavior of the dilatation for this family of braids. We also give an example of a family of pseudo-Anosov braids with arbitrarily small dilatation such that the mapping torus obtained from such braid has 2 cusps and has an arbitrarily large volume.
From pseudo-Anosov flows on graph manifolds to totally periodic flows
研究显示图流形上的伪Anosov流可通过构造部分Birkhoff截面，几乎等价于完全周期性伪Anosov流或悬垂Anosov流，结合先前工作证明图流形上的Transitive Anosov流几乎等价于悬垂Anosov流。
We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction of pseudo-Anosov maps to give explicit pseudo-Anosov maps of vanishing Sah-Arnoux-Fathi invariant. In particular, we give new infinite families of such maps in genus 3.
Finiteness of singular orbits of pseudo-Anosov flows
该研究证明闭合无环3维流形上伪Anosov流的奇异轨道对应的纽结同痕类数量有限，为相关几何拓扑问题提供了关键结论。
Generalized foliations for homeomorphisms isotopic to a pseudo-Anosov homeomorphism: a geometric realization of a result by Fathi
本研究给出Fathi一个结果的新几何证明，证明闭曲面上同痕于伪Anosov同胚的同胚都存在与伪Anosov同胚叶状结构性质相似的稳定、不稳定不变划分。
This paper concerns a family of pseudo-Anosov braids with dilatations arbitrarily close to one. The associated graph maps and train tracks have stable "star-like" shapes, and the characteristic polynomials of their transition matrices form Salem-Boyd sequences. These examples show that the logarithms of least dilatations of pseudo-Anosov braids on $2g+1$ strands are bounded above by $\log(2 + \sqrt{3})/g$. It follows that the asymptotic behavior of least dilatations of pseudo-Anosov, hyperelliptic surface homeomorphisms is identical to that found by Penner for general surface homeomorphisms. T…
We first prove rigidity results for pseudo-Anosov flows in prototypes of toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered manifold is up to finite covers topologically equivalent to a geodesic flow and we show that a pseudo-Anosov flow in a solv manifold is topologically equivalent to a suspension Anosov flow. Then we study the interaction of a general pseudo-Anosov flow with possible Seifert fibered pieces in the torus decomposition: if the fiber is associated with a periodic orbit of the flow, we show that there is a standard and very simple form for the flow in t…
Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity
本文提出一种利用伪Anosov流与veering三角剖分联系构造Birkhoff截面的新方法，可显式构造并控制Euler示性数与边界轨道复杂度，并证明任何传递伪Anosov流存在具有两个边界分量的Birkhoff截面。
Pseudo-Anosov flows and the geometry of Anosov-like group actions
研究伪Anosov流在闭3流形上的作用及其在Gromov超几何空间上的等距作用，揭示非R覆盖时群作用的弱proper discontinuous性质及周期轨道外元素的普遍性。
A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered Anosov flow. The proof uses the universal circle to R-covered foliations.
In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.
We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichmüller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a family of pseudo-Anosov maps which violates one side of inequalities under unbounded geometry setting, present an explicit bounding constant for a punctured torus, and provide several observations based on experiments.
The main result of this paper is a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms, ranging over all surfaces. More precisely, we consider pseudo-Anosovs F:S to S with |chi(S)| log(lambda(F)) bounded above by some constant, and we prove that, after puncturing the surfaces at the singular points of the stable foliations, the resulting set of mapping tori is finite. Said differently, there is a finite set of fibered hyperbolic 3-manifolds so that all small dilatation pseudo-Anosovs occur as the monodromy of a Dehn filling on one of the 3-manifolds in…
Let $F',F$ be any two closed orientable surfaces of genus $g'>g\ge 1$, and $f:F\to F$ be any pseudo-Anosov map. Then we can "extend" $f$ to be a pseudo-Anosov map $f':F'\to F'$ so that there is a fiber preserving degree one map $M(F',f')\to M(F,f)$ between the hyperbolic surface bundles. Moreover the extension $f'$ can be chosen so that the surface bundles $M(F',f')$ and $M(F,f)$ have the same first Betti numbers.
The Dynamics of Measurable Pseudo-Anosov Maps
本文研究可测伪阿诺索夫映射的动力学，证明其传递性、稠密周期点、敏感依赖性和遍历性，并提及球面上的非共轭例子族。
We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{2}. This is in contrast to Penner's asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surface of genus g tends to zero at rate 1/g (as g goes to infinity). We also show that our uniform lower bound sqrt{2} is sharp. More precisely the least dilatation of a pseudo-Anosov on a genus g>1 translation surface in a hyperelliptic component belongs to the interval ]sqrt{2},sqr…
An infinite family of generalized pseudo-Anosov homeomorphisms of the sphere S is constructed, and their invariant foliations and singular orbits are described explicitly by means of generalized train tracks. The complex strucure induced by the invariant foliations is described, and is shown to make S into a complex sphere. The generalized pseudo-Anosovs thus become quasiconformal automorphisms of the Riemann sphere, providing a complexification of the unimodal family which differs from that of the Fatou/Julia theory.
A characterization of pseudo-Anosov orbit spaces via bifoliated planes
通过双叶平面上的群作用，结合分支覆盖、转向三角剖分和新的紧性准则，刻画了可定向闭三维流形上可传递伪阿诺索夫流的轨道空间。
We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus $g \geqslant 2$, we show that there are positive constants $a_1 < a_2$ such that the minimal translation length is bounded below and above by $a_1/ g^2$ and $a_2/g^2$.
We show that a pseudo-Anosov map constructed as a product of the large power of Dehn twists of two filling curves always has a geodesic axis on the curve graph of the surface. We also obtain estimates of the stable translation length of a pseudo-Anosov map, when two filling curves are replaced by multicurves. Three main applications of our theorem are the following: (a) determining which word realizes the minimal translation length on the curve graph within a specific class of words, (b) giving a new class of pseudo-Anosov maps optimizing the ratio of stable translation lengths on the curve gr…
The minimum dilatation of pseudo-Anosov 5-braids is shown to be the largest zero $λ_5 \approx 1.72208$ of $x^4 - x^3 - x^2 - x + 1$ which is attained by $σ_1σ_2σ_3σ_4σ_1σ_2$.
Horizontal Goodman surgery and almost equivalence of pseudo-Anosov flows
本文推广了Goodman手术操作，证明对传递伪Anosov流进行水平Goodman手术可产生几乎等价的流，并得出相关等价与结构稳定性结果。
We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham's proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surface. For genus g=2 to 5, the mimimum dilatation is the smallest Salem number for polynomials of degree 2g.
Non-transitive pseudo-Anosov flows
本文研究了非转置伪Anosov流，通过群作用于轨道空间和无穷远处边界来恢复流的基本集和Smale顺序，并证明了流的转置性与周期轨道密度的关系。
Partial hyperbolicity and pseudo-Anosov dynamics
本文证明容许部分双曲微分同胚的双曲 3 维流形必容许 Anosov 流，并利用中心分支叶理和坍缩 Anosov 流完整分类相关微分同胚。
Irreducible train tracks for pseudo-Anosov homeomorphisms
本文提出了一种构造伪Anosov映射不可约列车轨道的方法，通过处理阻碍不可约性的分支来实现。
A filling curve $γ$ on a based surface $S$ determines a pseudo-Anosov homeomorphism $P(γ)$ of $S$ via the process of "point-pushing along $γ$." We consider the relationship between the self-intersection number $i(γ)$ of $γ$ and the dilatation of $P(γ)$; our main result is that the dilatation is bounded between $(i(γ)+1)^{1/5}$ and $9^{i(γ)}$. We also bound the least dilatation of any pseudo-Anosov in the point-pushing subgroup of a closed surface and prove that this number tends to infinity with genus. Lastly, we investigate the minimal entropy of any pseudo-Anosov homeomorphism obtained by pu…
Generalized Pseudo-Anosov Maps and Hubbard Trees
本研究建立二次多项式与曲面同胚动力学的联系，定义无交叉型哈伯德树，并证明此类树恰为可构造广义伪阿诺索夫映射的充要条件。
An algorithm is proposed that solves two decision problems for pseudo-Anosov elements in the mapping class group of a surface with at least one marked fixed point. The first problem is the root problem: decide if the element is a power and in this case compute the roots. The second problem is the symmetry problem: decide if the element commutes with a finite order element and in this case compute this element. The structure theorem on which this algorithm is based provides also a new solution to the conjugacy problem.
The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface S_g of genus g. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov homeomorphism of S_g tends to zero at the rate 1/g. We consider here the smallest dilatation of any pseudo-Anosov homeomorphism of S_g acting trivially on Gamma/Gamma_k, the quotient of Gamma = pi_1(S_g) by the k-th term of its lower central series, k > 0. In contrast to Penner's asymptotics, we prove that this minimal dilatation is bounded above and below, independen…
In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we constructed in a previous article. A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We als…
Quadratic-time computations for pseudo-Anosov mapping classes
提出一种二次时间算法，通过输入映射类群中表示伪Anosov映射类的单词，计算其拉伸因子和不变测度叶状结构。
A Teichmuller lattice is the orbit of a point in Teichmuller space under the action of the mapping class group. We show that the proportion of lattice points in a ball of radius r which are not pseudo-Anosov tends to zero as r tends to infinity. In fact, we show that if R is a subset of the mapping class group, whose elements have an upper bound on their translation length on the complex of curves, then proportion of lattice points in the ball of radius r which lie in R tends to zero as r tends to infinity.
Unimodal Measurable Pseudo-Anosov Maps
本文构造球面上连续变化的同胚族，每个映射为可测伪阿诺索夫映射，半共轭于帐篷映射，具有拓扑传递性、遍历性、稠密周期点及熵 log λ，且典型参数下湍流处处非正则。
Pseudo-Anosov Reeb flows and contact structures
本文提出伪Anosov接触结构的概念，证明其具有普遍紧致性和无 torsion 性，并探讨了其在伪Anosov流有限性猜想中的应用。
In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic unit whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$. As an application of the argument, we also obtain a statement on the number of closed geodesics of the same length in the moduli space of area one abelian differentials for low genus cases.
Let $δ_{g,n}$ be the minimal dilatation of pseudo-Anosovs defined on an orientable surface of genus $g$ with $n$ punctures. Tsai proved that for any fixed $g \ge 2$, the logarithm of the minimal dilatation $\log δ_{g,n}$ is on the order of $\frac{\log n}{n}$. The main result of this paper is that if $2g+1$ is relatively prime to $s$ or $s+1$ for each $0 \le s \le g$, then $$\limsup_{n \to \infty} \frac{n \log δ_{g,n}}{\log n} \le 2.$$
Pseudo-Anosov Flows: A Plane Approach
本文通过简化框架重新阐述伪Anosov流理论，结合1-2维动力学工具，提供高效的基础结果呈现。
A Smörgåsbord of (bi)contact structures, Reeb flows and pseudo-Anosov flows
本文为会议论文集撰写，围绕同一切触结构的两个不同Reeb流动力学关联问题，整理相关结果并提出更多问题与猜想，旨在邀请读者研究特定切触结构族。
Drilling veering triangulations with applications to pseudo-Anosov flows
本文开发了一种转向三角剖分钻孔算法，用于构造转向母三角剖分，应用于伪阿诺索夫流的闭轨道钻孔，为Ghys关于传递阿诺索夫流的猜想提供了计算证据。
Pseudo-Anosov surfaces and Dynamics in 3-manifolds
本文确定了闭可定向3-流形允许伪Anosov曲面的条件，并在不可约情形下证明自同胚同痕于新引入的“部分伪Anosov”同胚。
Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the set of all braid types Boyland's dynamical partial order can be defined. We describe the partial order on a family of braids and show that a period 3 orbit of pseudo-Anosov braid type implies the Smale-horseshoe map which is a factor possessing complicated chaotic dynamics.
In this paper, we develop a way to extract information about a random walk associated with a typical Thurston's construction. We first observe that a typical Thurston's construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston's construction that have finite second moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss …
Uniqueness of gluings and virtual finiteness of pseudo-Anosov flows on graph manifolds
本文刻画了粘合得到的伪Anosov流的轨道等价条件，结合相关描述证明了图流形上可递伪Anosov流有限性猜想的虚版本。
Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil
本文证明亏格二双曲纤维化纽结在非零分数德恩扭转系数下其伪阿诺索夫代表元必有不动点，从而证明纽结弗洛尔同调可检测五叶结，并给出相关同调理论的应用。
An Algorithmic Classification of Generalized Pseudo-Anosov Homeomorphisms via Geometric Markov Partitions
本文利用几何 Markov 分区的几何类型作为完全不变量，给出实现判据与判定算法，实现广义 pseudo-Anosov 同胚拓扑共轭类的算法分类。
Minimum dilatations of pseudo-Anosov braids
本文确定了足够大n值下n股伪Anosov辫的最小膨胀δn，证实了相关猜想，并基本解决了n穿孔球面上的最小膨胀问题。
Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity, hence the limit sets are continuous images of the circle. One important corollary is that if F is a Reebless finite depth foliation in a hyperbolic manifold, then it has the continuous extension property. Such finite depth foliations exist whenever the second Betti number is non zero. The result also applies to other classe…
On the set of normalized dilatations of fully-punctured pseudo-Anosov maps
本文改进全穿孔伪阿诺索夫映射环面 veering 三角剖分中四面体数量的上界，并结合计算方法确定归一化伸缩系数集合的最小元素为黄金比例的平方、最小聚点为其四次方。
This paper concerns the set $\hat{\mathcal{M}}$ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold $N$ by Dehn filling three cusps with a mild restriction. We prove that for each $g$ (resp. $g \not\equiv 0 \pmod{6}$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{\mathcal{M}}$ defined on a closed surface $\varSigma_g$ of genus $g$ is achieved by the monodromy of some $\varSigma_g$-bundle over the circle obtained from $N(\tfrac{3}{-2})$ or $N(\tfrac{1}{-2})$ by Dehn filling two cus…
Obstructing Anosov flows on cusped 3-manifolds
本文利用taut foliations与pseudo-Anosov flows的关系，找到了首批不支持veering triangulations的cusped hyperbolic 3-manifolds实例，证实了S. Schleimer的猜想。
Minimal stretch factors of orientation-reversing fully-punctured pseudo-Anosov maps
本文研究了反方向全穿伪Anosov映射的最小伸缩因子，证明了其与曲面欧拉特征的关系，并展示了该界限的渐近紧致性。
Entropy-minimizing diffeomorphisms of pseudo-Anosov type on K3 surfaces
本文在K3曲面上构造了无穷多个伪阿诺索夫型微分同胚，证明它们在同伦类中熵最小，且不保持任何复结构。
In the hyperspace of subcontinua of a compact surface we consider a second order Hausdorff distance. This metric space is compactified in such a way that the stable foliation of a pseudo-Anosov map is naturally identified with a hypercontinuum. We show that negative iterates of a stable arc converges to this hypercontinuum in the considered metric. Some dynamical properties of pseudo-Anosov maps, as topological mixing and the density of stable leaves, are generalized for cw-expansive homeomorphisms of pseudo-Anosov type on compact metric spaces.
We prove a classification theorem for transitive Anosov and pseudo-Anosov flows on closed 3-manifolds, up to orbit equivalence. In many cases, flows on a 3-manifold $M$ are completely determined by the set of free homotopy classes of their (unoriented) periodic orbits. The exceptional cases are flows with a special structure in their orbit space called a ``tree of scalloped regions"; in these cases the set of free homotopy classes of unoriented periodic orbits together with the additional data of a choice of sign for each $π_1(M)$-orbit of tree gives a complete invariant of orbit equivalence c…
Standardly embedded train tracks and pseudo-Anosov maps with minimum expansion factor
本文证明了全穿孔伪Anosov映射的扩张因子下界 $λ(f)^{|χ(S)|} \ge μ^4 \approx 6.85408$，其证明基于对标准嵌入火车轨道及其权重空间上Thurston辛形式的研究。
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