Agol cycles of pseudo-Anosov maps on the 2-punctured torus and 5-punctured sphere

A numerical investigation was carried out to study the mixing behavior of Stokes flows in a rectangular cavity stirred by three square rods. The square loops of the rods move in such a way that a pseudo-Anosov map can be built in the flow domain in the augmented phase space. The finite volume method was used, and the flow domain was meshed by staggered grids with the periodic boundary conditions of the rod motion being imposed by the mesh supposition technique. Fluid particle tracking was carried out by a fourth-order Runge–Kutta scheme. Tracer stretches from different initial positions were used to evaluate interface prediction by a pseudo-Anosov map. The colored short period Poincaré section was obtained to reveal the size of the domain in which the pseudo-Anosov map was in effect. Dye advection patterns were used to analyze chaotic advection of passive tracer particles using statistical concepts such as “variances” and “complete spatial randomness.” For the fluid in the core region of the cavity, tracer interface stretches experienced exponential increases and had the same power index as that predicted by the pseudo-Anosov map matrix.

Starting from any pseudo-Anosov map ϕ on a surface of genus g⩾2 , we construct explicitly a family of Derived from pseudo-Anosov maps f by adapting the construction of Smale’s derived from Anosov maps on the two-torus. This is done by perturbing ϕ at some fixed points. We first consider perturbations at every conical fixed point and then at regular fixed points. We establish the existence of a measure µ, supported by the non-trivial unique minimal component of the stable foliation of f, with respect to which f is mixing. In the process, we construct a uniquely ergodic generalized interval exchange transformation (GIET) with a wandering interval that is semi-conjugated to a self-similar interval exchange transformation. This GIET is obtained as the Poincaré map of a flow renormalized by f which parametrizes stable foliation. When f is , the flow and the GIET are .

Let \( \tilde S \) be a Riemann surface of analytically finite type (p, n) with 3p − 3 + n > 0. Let a ∈ \( \tilde S \) and S = \( \tilde S \) − {a}. In this article, the author studies those pseudo-Anosov maps on S that are isotopic to the identity on \( \tilde S \) and can be represented by products of Dehn twists. It is also proved that for any pseudo-Anosov map f of S isotopic to the identity on \( \tilde S \), there are infinitely many pseudo-Anosov maps F on S − {b} = \( \tilde S \) − {a, b}, where b is a point on S, such that F is isotopic to f on S as b is filled in.

Let $\tilde{S}$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $F$ be a pseudo-Anosov map of $\tilde{S}$ defined by two filling simple closed geodesics on $\tilde{S}$. Let $a\in \tilde{S}$, and $S=\tilde{S} - \{a\}$. For any map $f\colon S\to S$ that is generated by two simple closed geodesics and is isotopic to $F$ on $\tilde{S}$, there corresponds to a configuration $\tau$ of invariant half planes in the universal covering space of $\tilde{S}$. We give a necessary and sufficient condition (with respect to the configuration) for those $f$ to be pseudo-Anosov maps. As a consequence, we obtain infinitely many pseudo-Anosov maps $f$ on $S$ that are isotopic to $F$ on $\tilde{S}$ as $a$ is filled in.

Let M be a closed oriented surface of genus at least two, and let SF denote the space of all projective classes of measured foliations on M. The authors have previously given a criterion in terms of certain combinatorial words for an element of &^ to be left invariant by a pseudo-Anosov map of M: such foliations are characterized by the fact that the associated word is eventually periodic. The current work derives an estimate which says roughly that the dilatation of the corresponding pseudo-Anosov map is large if the periodic part of the word is long. This estimate is then used to bound the number of distinct conjugacy classes of foliations invariant under pseudo-Anosov maps of M in terms of a specified bound on the dilatations. 1. Introduction. This paper is a sequel to our paper [5] in which we described a method of representing measured foliations carried by a train track by means of semi-infinite combinatorial words in a finite alphabet. We showed that pseudo-Anosov foliations (i.e., those that are preserved by pseudo-Anosov maps) are characterized by being representable by eventually periodic convergent words. In this paper, we establish an inequality relating the length of the repeating part of the word corresponding to a pseudo-Anosov foliation and the dilatation factor of a pseudo-Anosov map preserving that foliation. As a by-product, for each real number R, we shall describe a finite set of foliations, whose cardinality is bounded in terms of R and which contains a representative of each conjugacy class of pseudo-Anosov foliation which is preserved by a pseudo-Anosov map of dilatation at most R. Throughout this paper, M will denote a closed, oriented, smooth surface of negative Euler characteristic. Moreover, unless otherwise stated, we will tacitly assume that each train track considered is trivalent, i.e., there are exactly three half-branches incident on each switch. For the basic facts about measured foliations and train tracks, we refer the reader to [1], [3], [4], [5], or [6]. It is necessary to further refine some of the notions developed in [5] (whose terminology we will adopt), and this is the purpose of the next paragraph.

We view braids as automorphisms of punctured disks and define a partial order on pseudo-Anosov braids called the “forcing order”. The order measures whether one automorphism induces another given automorphism on the surface. Pseudo-Anosov growth rate decreases relative to the order and appears to give a good measure of braid complexity. Unfortunately it appears difficult computationally to determine explicitly the partial order structure by hand. We use several computer algorithms to study the bottom part of the partial order when the number of braid strands is fixed. From the algorithms, we build sequences of low entropy pseudo-Anosov n-strand braids that are minimal in the sense that they do not force any other pseudo-Anosov braids on the same number of strands. The sequences are an extension of work done by Hironaka and Kin, and we conjecture the sequences to achieve minimal entropy among certain non-trivial classes of braids. In general, the lowest entropy pseudo-Anosov braids appear to have mapping tori that come from Dehn surgery on very low volume hyperbolic 3-manifolds, and we begin to analyze the relation between entropy and hyperbolic volume. Moreover, the low-growth families contain non-trivial low-growth families of horseshoe braids and we proceed to study dynamics of the horseshoe map as well.

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.

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 unit circle. This shows that the techniques that were used to disprove Penner's conjecture on orientable surfaces are ineffective in the nonorientable cases.

Fast dynamo models based on cat maps with shear are introduced. With an appropriate choice of shear even and odd magnetic fields evolve independently. Fast dynamo action occurs for even fields in the absence of shear; the introduction of shear introduces cancellations and modifies growth rates. An odd field may be considered as lying in a disc and evolving under a pseudo-Anosov map, which stretches and folds field but does not reconnect field lines. Without shear odd fields decay, but non-trivial shear allows fast amplification of field by the stretch–fold–shear mechanism. Numerical evidence is presented showing that these models can act as fast dynamos for both even and odd fields. The limit of large stretching by the cat map is considered and proofs of fast dynamo action in this limit are presented.

Let $S$ be a Riemann surface containing at least two punctures $z$ and $z_0$. Let $\mathscr{F}(S)$ be the set of pseudo-Anosov maps of $S$ that are isotopic to the identity on $S\cup \{z\}$. We show that for any $f\in \mathscr{F}(S)$ and any twice punctured disk $\Delta$ enclosing $z$ and $z_0$, the pair $(\partial \Delta, f(\partial \Delta))$ fills $S$, where $\partial \Delta$ denotes the boundary of $\Delta$. Fix such a $\Delta$, and denote by $\mathscr{T}(\Delta)$ the set of twice punctured disks $\Delta'$ on $S$ enclosing $z$ and $z_0$ with the property that $(\partial \Delta, \partial \Delta')$ fills $S$. Let $\Delta_0\in \mathscr{T}(\Delta)$. We describe all possible pseudo-Anosov maps $f$ in $\mathscr{F}(S)$ sending $\Delta$ to $\Delta_0$, and classify elements of $\mathscr{F}(S)$ in terms of $\mathscr{T}(\Delta)$. We also show that there are infinitely many elements $f_k\in \mathscr{F}(S)$ with $f_k(\Delta)=\Delta_0$ such that their dilatations $\lambda(f_k)\rightarrow +\infty$ as $k\rightarrow +\infty$.

Let F′,F be any two closed orientable surfaces of genus g′ > g≥ 1, and f:F→ F be any pseudo-Anosov map. Then we can “extend” f to be a pseudo- Anosov map f′:F′→ F′ so that there is a fiber preserving degree one map M(F′,f′)→ 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.

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.

Measured Foliations

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

Measured foliations