pseudo-Anosov Homeomorphism Research Articles

Constructing pseudo-Anosovs from expanding interval maps

We investigate a phenomenon observed by 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 \phi_g on the closed surface of genus g that preserves an algebraically primitive translation structure and whose dilatation is a Salem number.

Pseudo-Anosov homeomorphisms of punctured nonorientable surfaces with small stretch factor

Pseudo-Anosov homeomorphisms of punctured nonorientable surfaces with small stretch factor

A construction of pseudo-Anosov homeomorphisms using positive twists

A construction of pseudo-Anosov homeomorphisms using positive twists

Hamiltonian flows for pseudo-Anosov mapping classes

For a given pseudo-Anosov homeomorphism \varphi of a closed surface S , the action of \varphi on the Teichmüller space \mathcal{T}(S) preserves the Weil–Petersson symplectic form. We give explicit formulae for two invariant functions \mathcal{T}(S)\to \mathbb{R} whose symplectic gradients generate autonomous Hamiltonian flows that coincide with the action of \varphi at time one. We compute the Poisson bracket between these two functions. This amounts to computing the variation of length of a Hölder cocycle on one lamination along a shear vector field defined by another. For a measurably generic set of laminations, we prove that the variation of length is expressed as the cosine of the angle between the two laminations integrated against the product Hölder distribution, generalizing a result of Kerckhoff. We also obtain rates of convergence for the supports of germs of differentiable paths of measured laminations in the Hausdorff metric on a hyperbolic surface, which may be of independent interest.

Geodesics in the mapping class group

We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism f, a sequence of points w_k and a sequence of radii r_k so that the ball B(w_k, r_k) is disjoint from a quasi-axis a of f, but for any projection map from mapping class group to a, the diameter of the image of B(w_k, r_k) grows like log(r_k).

Lengths spectrum of hyperelliptic components

We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmüller geodesic flow restricted to the hyperelliptic connected components of the strata of Abelian differentials, settling a question of Farb [Problems on Mapping Class Groups and Related Topics, 11–55 (2006)]. These are the first results on the description of the systole of moduli spaces, outside some cases in low genera. We emphasize that all proofs and computations can be performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components, and we provide a picture of that for small genera.

The Geometry of Bi-Perron Numbers with Real or Unimodular Galois Conjugates

Abstract Among all bi-Perron numbers, we characterise those all of whose Galois conjugates are real or unimodular as the ones that admit a power that is the stretch factor of a pseudo-Anosov homeomorphism arising from Thurston’s construction. This is in turn equivalent to admitting a power that is the spectral radius of a bipartite Coxeter transformation.

Limits of sequences of pseudo-Anosov maps and of hyperbolic 3–manifolds

There are two objects naturally associated with a braid $\beta\in B_n$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism $\varphi_\beta\colon S^2\to S^2$; and the finite volume complete hyperbolic structure on the 3-manifold $M_\beta$ obtained by excising the braid closure of $\beta$, together with its braid axis, from $S^3$. We show the disconnect between these objects, by exhibiting a family of braids $\{\beta_q:q\in{\mathbb{Q}}\cap(0,1/3]\}$ with the properties that: on the one hand, there is a fixed homeomorphism $\varphi_0\colon S^2\to S^2$ to which the (suitably normalized) homeomorphisms $\varphi_{\beta_{q}}$ converge as $q\to 0$; while on the other hand, there are infinitely many distinct hyperbolic 3-manifolds which arise as geometric limits of the form $\lim_{k\to\infty} M_{\beta_{q_k}}$, for sequences $q_k\to 0$.

Realization of homotopy classes of torus homeomorphisms by the simplest structurally stable diffeomorphisms

Abstract. According to Thurston’s classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. A homotopy class from each subset is characterized by the existence of a homeomorphism called Thurston’s canonical form, namely: a periodic homeomorphism, a reducible nonperiodic homeomorphism of algebraically finite order, a reducible homeomorphism that is not a homeomorphism of an algebraically finite order, and a pseudo-Anosov homeomorphism. Thurston’s canonical forms are not structurally stable diffeomorphisms. Therefore, the problem naturally arises of constructing the simplest (in a certain sense) structurally stable diffeomorphisms in each homotopy class. In this paper, the problem posed is solved for torus homeomorphisms. In each homotopy class, structurally stable representatives are analytically constructed, namely, a gradient-like diffeomorphism, a Morse-Smale diffeomorphism with an orientable heteroclinic, and an Anosov diffeomorphism, which is a particular case of a pseudo-Anosov diffeomorphism.

Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs

In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds. Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichm\uller to curve graph translation length are, in a reasonable sense, ubiquitous in the connected components of strata of Abelian differentials. Finally, we present a further application to filling pairs on punctured surfaces by constructing filling pairs whose algebraic and geometric intersection numbers are equal.

Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

AbstractWe develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection

According to the Nielsen – Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: $T_{1}$) periodic homeomorphism; $T_{2}$) reducible non-periodic homeomorphism of algebraically finite order; $T_{3}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_{4}$) pseudo-Anosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are $T_{1}$, $T_{2}$, $T_{4}$ only. Moreover, all representatives of the class $T_{4}$ have chaotic dynamics, while in each homotopy class of types $T_{1}$ and $T_{2}$ there are regular diffeomorphisms, in particular, Morse – Smale diffeomorphisms with a finite number of heteroclinic orbits. The author has found a criterion that allows one to uniquely determine the homotopy type of a Morse – Smale diffeomorphism with a finite number of heteroclinic orbits on a two-dimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and non-trivial. It is proved that if the heteroclinic points of a Morse – Smale diffeomorphism are contained only in the trivial domains then such diffeomorphism has the homotopic type $T_{1}$. The orbit space of non-trivial heteroclinic domains consists of a finite number of two-dimensional tori, where the saddle separatrices participating in heteroclinic intersections are projected as transversally intersecting knots. That whether the Morse – Smale diffeomorphisms belong to types $T_{1}$ or $T_{2}$ is uniquely determined by the total intersection index of such knots.

Diffeomorphisms of 2-manifolds with one-dimensional spaciously situated basic sets

We consider orientation-preserving -diffeomorphisms of orientable surfaces of genus greater than one with a one-dimensional spaciously situated perfect attractor. We show that the topological classification of restrictions of diffeomorphisms to such basic sets can be reduced to that of pseudo-Anosov homeomorphisms with a distinguished set of saddles. In particular, we prove a result announced by Zhirov and Plykin, which gives a topological classification of the -diffeomorphisms of the surfaces under discussion under the additional assumption that the non-wandering set consists of a one-dimensional spaciously situated attractor and zero-dimensional sources.

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.

Classification of one-dimensional attractors of diffeomorphisms of surfaces by means of pseudo-anosov homeomorphisms

In the present paper axiom diffeomorphisms of closed 2-manifolds of genus whose nonwandering set contains perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of the paper is the following. Two diffeomorphisms from the given class are topologically conjugate on attractors if and only if corresponding pseudo-Anosov homeomorphisms are topologically conjugate by means of homeomorphism that maps a certain subset of one pseudo-Anosov map onto the certain subset of the other pseudo-Anosov map.

An algorithm to compute the Teichmüller polynomial from matrices

In their precedent work, the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov homeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithm to compute the Teichmuller polynomial corresponding to those surface homeomorphisms by first constructing an invariant track whose first homology group can be naturally identified with the first homology group of the surface, and computing its Alexander polynomial.

The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups

We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In an appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichmüller translation distance.

Classification of One-Dimensional Attractors of Diffeomorphisms of Surfaces by Means of Pseudo-Anosov Homeomorphisms

Classification of One-Dimensional Attractors of Diffeomorphisms of Surfaces by Means of Pseudo-Anosov Homeomorphisms

An infinite surface with the lattice property Ⅱ: Dynamics of pseudo-Anosovs

We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfies a mixing-type formula with polynomial decay. Then we consider the extent to which the action of these hyperbolic affine automorphisms satisfy Thurston's definition of a pseudo-Anosov homeomorphism. In particular we study the action of these automorphisms on simple closed curves and on homology classes. These objects are exponentially attracted by the expanding and contracting foliations but exhibit polynomial decay. We are able to work out exact asymptotics of these limiting quantities because of special integral formula for algebraic intersection number which is attuned to the geometry of the surface and its deformations.

Any Baumslag–Solitar action on surfaces with a pseudo-Anosov element has a finite orbit

We consider homeomorphisms$f,h$generating a faithful$\mathit{BS}(1,n)$-action on a closed surface$S$, that is,$hfh^{-1}=f^{n}$for some$n\geq 2$. According to Guelman and Liousse [Actions of Baumslag–Solitar groups on surfaces.Discrete Contin. Dyn. Syst. A 5(2013), 1945–1964], after replacing$f$by a suitable iterate if necessary, we can assume that there exists a minimal set$\unicode[STIX]{x1D6EC}$of the action, included in$\text{Fix}(f)$. Here, we suppose that$f$and$h$are$C^{1}$in a neighborhood of$\unicode[STIX]{x1D6EC}$and any point$x\in \unicode[STIX]{x1D6EC}$admits an$h$-unstable manifold$W^{u}(x)$. Using Bonatti’s techniques, we prove that either there exists an integer$N$such that$W^{u}(x)$is included in$\text{Fix}(f^{N})$or there is a lower bound for the norm of the differential of$h$depending only on$n$and the Riemannian metric on $S$. Combining the last statement with a result of Alonso, Guelman and Xavier [Actions of solvable Baumslag–Solitar groups on surfaces with (pseudo)-Anosov elements.Discrete Contin. Dyn. Syst.to appear], we show that any faithful action of$\mathit{BS}(1,n)$on$S$with$h$a pseudo-Anosov homeomorphism has a finite orbit containing singularities of $h$; moreover, if$f$is isotopic to the identity, it is entirely contained in the singular set of $h$. As a consequence, there is no faithful$C^{1}$-action of$\mathit{BS}(1,n)$on the torus with$h$Anosov.