A note on pseudo-Anosov maps with small growth rate

A family of pseudo-Anosov braids with large conjugacy invariant sets

We show that there is a family of pseudo-Anosov braids independently parameterized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length and conclude that the conjugacy problem remains exponential in the braid index under the current knowledge.

A geometric criterion to be pseudo-Anosov

We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions. Specifically, any element of the fundamental group of a surface S gives rise to a mapping class on the punctured surface, and we show that such a class is pseudo-Anosov if its geodesic representative is "wide" in some hyperbolic 3-manifold homeomorphic to the trivial interval bundle over S.

环面上曲面丛的伪 Anosov 子群

Pseudo-Anosov subgroups of surface bundles over tori

本文利用 Birman 正合列证明环面上曲面丛基本群中的有限生成纯伪 Anosov 子群在映射类群中凸余紧，从而推广纤维化三维流形群的相关结论。

A positive proportion of elements of mapping class groups is pseudo-Anosov

In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.

一般纤维化3-流形群的伪Anosov子群

Pseudo-Anosov subgroups of general fibered 3-manifold groups

本文证明具有可约化单项式的纤维化3-流形中有限生成纯伪Anosov子群经Birman正合序列在映射类群中是凸余紧的，并结合已有结果确立了所有此类群像的结论。

Subgroups generated by two pseudo-Anosov elements in a mapping class group. II. Uniform bound on exponents

Let $S$ be a compact orientable surface, and $\Mod(S)$ its mapping class group. Then there exists a constant $M(S)$, which depends on $S$, with the following property. Suppose $a,b \in \Mod(S)$ are independent (i.e., $[a^n,b^m]\not=1$ for any $n,m \not=0$) pseudo-Anosov elements. Then for any $n,m \ge M$, the subgroup $<a^n,b^m>$ is free of rank two, and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in $<a^n,b^m>$ are pseudo-Anosov. We also show that there exists a constant $N$, which depends on $a,b$, such that $<a^n,b^m>$ is free of rank two and convex-…

Agol cycles of pseudo-Anosov 3-braids

An Agol cycle is a complete invariant of the conjugacy class of a pseudo-Anosov mapping class. We study necessary and sufficient conditions for equivalent Agol cycles of pseudo-Anosov 3-braids.

3-流形中的伪Anosov曲面与动力学

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。

A homological recipe for pseudo-Anosovs

We give a simple explicit construction of pseudo-Anosov mapping classes using an improvement of the homological criterion of Casson-Bleiler.

将伪阿诺索夫映射类计数为弱收缩等距映射

Counting pseudo-Anosovs as weakly contracting isometries

本文证明在有限型双曲曲面映射类群及更广的层级双曲群的Cayley图中，伪阿诺索夫映射类或Morse元素是普遍的，并建立了拟等距不变量计数理论。

On the minimum of asymptotic translation lengths of point-pushing pseudo-Anosov maps on one punctured Riemann surfaces

We show that the minimum of asymptotic translation lengths of all point-pushing pseudo-Anosov maps on any one punctured Riemann surface is one.

The kernel of $Burau(4) \otimes Z_p$ is all pseudo-Anosov

We show that the kernel of $Burau(4) \otimes Z_p$, the reduced Burau representation with coefficients in $Z_p$ of the 4-braid group $B_4$, consists only of pseudo-Anosov braids.

Genericity of pseudo-Anosov mapping classes, when seen as mapping classes

We prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes.

On genericity of pseudo-Anosovs in the Torelli group

We show that, for any (symmetric) finite generating set of the Torelli group of a closed surface, the probability that a random word is not pseudo-Anosov decays exponentially in terms of the length of the word.

Pseudo-Anosov homeomorphisms not arising from branched covers

In this paper we provide a negative answer to a question of Farb about the relation between the algebraic degree of the stretch factor of a pseudo-Anosov homeomorphism and the genus of the surface on which it is defined.

Invariant geodesics in the curve complex under point-pushing pseudo-Anosov mapping classes

Let $S$ be a closed Riemann surface of genus $p>1$ with one point removed. In this paper, we identify those point-pushing pseudo-Anosov maps on $S$ that preserve at least one bi-infinite geodesic in the curve complex.

Pseudo-Anosov mapping classes not arising from Penner's construction

We show that Galois conjugates of stretch factors of pseudo-Anosov mapping classes arising from Penner's construction lie off the unit circle. As a consequence, we show that for all but a few exceptional surfaces, there are examples of pseudo-Anosov mapping classes so that no power of them arises from Penner's construction. This resolves a conjecture of Penner.

伪阿诺索夫映射类在映射类群中指数级地普遍

Pseudo-Anosovs are exponentially generic in mapping class groups

本文证明在映射类群中，通过适当选取有限生成集，非伪阿诺索夫映射类在半径球中的比例指数级趋于零，且方法适用于弱双曲群。

Small dilatation pseudo-Anosov mapping classes and short circuits on train track automata

This note gives a brief survey of the minimum dilatation problem for pseudo-Anosov mapping classes, and the first explicit train track description of an infinite family of pseudo-Anosov mapping classes with orientable stable foliations and the conjectural minimum dilatation for closed surfaces of even genus $g \ge 2$.

Pseudo-Anosov dilatations and the Johnson filtration

Answering a question of Farb-Leininger-Margalit, we give explicit lower bounds for the dilatations of pseudo-Anosov mapping classes lying in the kth term of the Johnson filtration of the mapping class group.

Virtual homological eigenvalue and mapping torus of pseudo-Anosov maps

In this note, we show that, if a pseudo-Anosov map $ϕ:S\to S$ admits a finite cover whose action on the first homology has spectral radius greater than $1$, then the monodromy of any fibered structure of any finite cover of the mapping torus $M_ϕ$ has the same property.

Galois conjugates of pseudo-Anosov stretch factors are dense in the complex plane

In this paper, we study the Galois conjugates of stretch factors of pseudo-Anosov elements of the mapping class group of a surface. We show that - except in low-complexity cases - these conjugates are dense in the complex plane. For this, we use Penner's construction of pseudo-Anosov mapping classes. As a consequence, we obtain that in a sense there is no restriction on the location of Galois conjugates of stretch factors arising from Penner's construction. This complements an earlier result of Shin and the author stating that Galois conjugates of stretch factors arising from Penner's construc…

A Farey tree structure on a family of pseudo-Anosovs

We introduce a new perspective on a procedure for generating pseudo-Anosov homemorphisms from postcritically finite interval maps. The central idea is the realization of a tree structure on one such family of pseudo-Anosovs: individual systems, represented by the rational number measuring their rotation at infinity, are the vertices of the tree, while the edges encode dynamical relations between them. We also deepen the dictionary between one-, two- and three-dimensional invariants associated with these systems.

Extending powers of pseudo-Anosovs

Biringer, Johnson, and Minsky showed that a pseudo-Anosov map on a boundary component of an irreducible 3-manifold has a power that partially extends to the interior if and only if the (un)stable laminations of $f$ is an $\mathbb{R}$-projective limit of meridians. We prove that the power required for a pseudo-Anosov map to partially extend is not universally bounded. We construct a family of pseudo-Anosov maps $f_i$ for all $i=1,2...$ on a boundary component of a family of irreducible 3-manifolds $M_i$ such that $f_i^i$ partially extends to the interior of $M_i$ but $f_i^j$ does not for $j<i$.

Minimal pseudo-Anosov stretch factors on nonoriented surfaces

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliations on the closed orientable surfaces of genus 1, 3, 5, 7, 9 and 11. As a byproduct, we obtain that the stretch factor of a pseudo-Anosov map on a nonorientable surface or an orientation-reversing pseudo-Anosov map on an orientable surface does not have Galois conjugates on the uni…

The Number of Pseudo-Anosov Elements in the Mapping Class Group of a Four-Holed Sphere

We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the identity tends to one as the radius of the ball tends to infinity.

Commensurability of two-multitwist pseudo-Anosovs

This paper analyzes commensurability of the class of surface automorphism generated by two Dehn multitwists. We show pairwise noncommensurability between several classes arising from canonical curve configurations. In addition, we consider the Kenyon-Smillie invariant J of flat surfaces in this setting. We also introduce a general construction of infinite classes of commensurable pseudo-Anosov homeomorphisms.

Hyperbolic structures from Sol on pseudo-Anosov mapping tori

The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov $ϕ:S\rightarrow S$ has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on the closed surface, then the Sol structure can be deformed to nearby cone hyperbolic structures, in the sense of projective structures. The cone angles can be chosen to be decreasing from multiples of $2π$.

Low dilatation pseudo-Anosovs on punctured surfaces and volume

For a pseudo-Anosov homeomorphism $f$ on a closed surface of genus $g\geq 2$, for which the entropy is on the order $\frac{1}{g}$ (the lowest possible order), Farb-Leininger-Margalit showed that the volume of the mapping torus is bounded, independent of $g$. We show that the analogous result fails for a surface of fixed genus $g$ with $n$ punctures, by constructing pseudo-Anosov homeomorphism with entropy of the minimal order $\frac{\log n}{n}$, and volume tending to infinity.

Pseudo-Anosov subgroups of fibered 3-manifold groups

Let X be a hyperbolic surface and H the fundamental group of a hyperbolic 3-manifold that fibers over the circle with fiber X. Using the Birman exact sequence, H embeds in the mapping class group Mod(Y) of the surface Y obtained by removing a point from X. We prove that a subgroup G in H is convex cocompact in Mod(Y) if and only if G is finitely generated and purely pseudo-Anosov. We also prove a generalization of this theorem with H replaced by an arbitrary Gromov hyperbolic extension of the fundamental group of X, and an additional hypothesis of quasi-convexity of G in H. Along the way, we o…

On the flux of pseudo-Anosov homeomorphisms

We exhibit a pseudo-Anosov homeomorphism of a surface S which acts trivially on the first homology group of S and whose flux is non zero

Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant

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.

利用正半扭转构造伪阿诺索夫同胚

A construction of pseudo-Anosov homeomorphisms using positive twists

本文提出一种仅用正半扭转构造穿孔球面及高亏格曲面上伪阿诺索夫同胚的方法，可生成拉伸因子具有特定数论性质（如迹域非全实、伽罗瓦共轭在单位圆上）的显式例子。

Extending pseudo-Anosov maps to compression bodies

We show that a pseudo-Anosov map on a boundary component of an irreducible 3-manifold has a power that partially extends to the interior if and only if its (un)stable lamination is a projective limit of meridians. The proof is through 3-dimensional hyperbolic geometry, and involves an investigation of algebraic limits of convex cocompact compression bodies.

伪Anosov映射的固定点

On fixed points of pseudo-Anosov maps

本文提出估计伪Anosov映射固定点数量的公式，并探讨其在Heegaard Floer同调秩与映射扭体超广义体积之间的不等式应用。

从拟测地流到伪Anosov流

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。

Sequences of pseudo-Anosov mapping classes and their asymptotic behavior

We construct sequences of pseudo-Anosov mapping classes whose dilatations behave asymptotically like the inverse of the Euler characteristic of the surface they are defined on. These sequences are used to show that if the genus, g, and punctures, n, of a surface are related by a rational ray g=rn then the minimal dilatations behave asymptotically like the inverse of the Euler characteristic.

Pseudo-Anosov maps and simple closed curves on surfaces

Given a pair of curves C_1 and C_2 on a hyperbolic surface F, when does there exist a pseudo-Anosov map sending one to another? More generally, one may ask the same question for C_i to be sets of disjoint simple closed curves. We will give necessary and sufficient conditions for the existence of such maps.

A recipe for short-word pseudo-Anosovs

Given any generating set of any pseudo-Anosov-containing subgroup of the mapping class group of a surface, we construct a pseudo-Anosov with word length bounded by a constant depending only on the surface. More generally, in any subgroup G we find an element f with the property that the minimal subsurface supporting a power of f is as large as possible for elements of G; the same constant bounds the word length of f. Along the way we find new examples of convex cocompact free subgroups of the mapping class group.

Infinite sequence of fixed point free pseudo-Anosov homeomorphisms

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.

Constructing pseudo-Anosovs from expanding interval maps

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…

伪Anosov流、双曲几何与曲线图

Pseudo-Anosov flows, hyperbolic geometry, and the curve graph

本文研究了伪Anosov流与双曲几何之间的关系，通过曲线图将流的动力学特性与几何特征联系起来。

曲面上伪 Anosov 映射的度量弱真不连续性

Metric weak proper discontinuity for pseudo-Anosov maps

提出伪 Anosov 映射的度量版弱真不连续性，并证明环面与双曲曲面同胚群上存在在具大不动点集同胚上无界的拟态。

Ideal Triangulations of Pseudo-Anosov Mapping Tori

We show how to construct an ideal triangulation of a mapping torus of a pseudo-Anosov map punctured along the singular fibers. This gives rise to a new conjugacy invariant of mapping classes, and a new proof of a theorem of Farb-Leininger-Margalit. The approach in this paper is based on ideas of Hamenstadt.

Some pseudo-Anosov maps on punctured Riemann surfaces represented by multi-twists

This paper has been withdrawn by the author

Extensions, quotients and generalized pseudo-Anosov maps

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.

A new relationship between the dilatation of pseudo-Anosov braids and fixed point theory

A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this paper we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to B. Jiang and H. Zheng.

An asymptotic behavior of the dilatation for a family of pseudo-Anosov braids

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.

The asymptotic behavior of the minimal pseudo-Anosov dilatations in the hyperelliptic handlebody groups

We consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the handlebody. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of…

从图流形上的伪Anosov流到完全周期性流

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。

On the genericity of pseudo-Anosov braids II: conjugations to rigid braids

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n $\ge$ 3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated "easily" into a rigid braid.

On pseudo-Anosov maps with small dilatations on punctured Riemann spheres

Let $S_n$ be a punctured Riemann spheres $\mathbf{S}^2\backslash \{x_1,..., x_n\}$. In this paper, we investigate pseudo-Anosov maps on $S_n$ that are isotopic to the identity on $S_n\cup \{x_n\}$ and have the smallest possible dilatations. We show that those maps cannot be obtained from Thurston's construction (that is the products of two Dehn twists). We also prove that those pseudo-Anosov maps $f$ on $S_n$ with the minimum dilatations can never define a trivial mapping class as any puncture $x_i$ of $S_n$ is filled in. The main tool is to give both lower and upper bounds estimations for dil…

New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant

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.

伪 Anosov 3-辫子的 Agol 循环的完整描述

Complete description of Agol cycles of pseudo-Anosov 3-braids

本文通过研究 Mosher 与 Farey 区间相关的 train tracks，完整刻画了所有伪 Anosov $3$-辫子的 Agol 循环。