arXiv subjects

Toby Hall

Publications and source records attributed to Toby Hall.

At least 19 recordsLinked to original sources

arXiv2023

The Dynamics of Measurable Pseudo-Anosov Maps

We study the dynamics of measurable pseudo-Anosov homeomorphisms of surfaces, a generalization of Thurston's pseudo-Anosov homeomorphisms. A measurable pseudo-Anosov map has a transverse pair of full measure turbulations consisting of streamlines which are dense immersed lines: these turbulations are equipped with measures which are expanded and contracted uniformly by the homeomorphism. The turbulations need not have a good product structure anywhere, but have some local structure imposed by the existence of tartans: bundles of unstable and stable streamline segments which intersect regularly, and on whose intersections the product of the measures on the turbulations agrees with the ambient measure. We prove that measurable pseudo-Anosov maps are transitive, have dense periodic points, sensitive dependence on initial conditions, and are ergodic with respect to the ambient measure. Measurable pseudo-Anosovs maps were introduced in [11], where we constructed a parameterized family of non-conjugate examples on the sphere.

math.DS↗

arXiv2023

Unimodal Measurable Pseudo-Anosov Maps

We exhibit a continuously varying family $F_\lambda$ of homeomorphisms of the sphere $S^2$, for which each $F_\lambda$ is a measurable pseudo-Anosov map. Measurable pseudo-Anosov maps are generalizations of Thurston's pseudo-Anosov maps, and also of the generalized pseudo-Anosov maps of [19]. They have a transverse pair of invariant full measure turbulations, consisting of streamlines which are dense injectively immersed lines: these turbulations are equipped with measures which are expanded and contracted uniformly by the homeomorphism. The turbulations need not have a good product structure anywhere, but have some local structure imposed by the existence of tartans: bundles of unstable and stable streamline segments which intersect regularly, and on whose intersections the product of the measures on the turbulations agrees with the ambient measure. Each map $F_\lambda$ is semi-conjugate to the inverse limit of the core tent map with slope $\lambda$: it is topologically transitive, ergodic with respect to a background Oxtoby-Ulam measure, has dense periodic points, and has topological entropy $h(F_\lambda) = \log \lambda$ (so that no two $F_\lambda$ are topologically conjugate). For a full measure, dense $G_\delta$ set of parameters, $F_\lambda$ is a measurable pseudo-Anosov map but not a generalized pseudo-Anosov map, and its turbulations are nowhere locally regular.

math.DS↗

arXiv2019

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 $q_k\to 0$.

math.GT↗

arXiv2018

Statistical Stability for Barge-Martin attractors derived from tent maps

Let $\{f_t\}_{t\in(1,2]}$ be the family of core tent maps of slopes $t$. The parameterized Barge-Martin construction yields a family of disk homeomorphisms $\Phi_t\colon D^2\to D^2$, having transitive global attractors $\Lambda_t$ on which $\Phi_t$ is topologically conjugate to the natural extension of $f_t$. The unique family of absolutely continuous invariant measures for $f_t$ induces a family of ergodic $\Phi_t$-invariant measures $\nu_t$, supported on the attractors $\Lambda_t$. We show that this family $\nu_t$ varies weakly continuously, and that the measures $\nu_t$ are physical with respect to a weakly continuously varying family of background Oxtoby-Ulam measures $\rho_t$. Similar results are obtained for the family $\chi_t\colon S^2\to S^2$ of transitive sphere homeomorphisms, constructed in [17] as factors of the natural extensions of $f_t$.

math.DS↗

arXiv2017

Typical path components in tent map inverse limits

In the inverse limit ${\hat{I}}_s$ of a tent map $f_s$ restricted to its core, the set $\mathcal{GR}$ of points whose path components are bi-infinite and bi-dense has full measure with respect to the measure induced on $\hat{I}_s$ by the unique absolutely continuous invariant measure of $f_s$. With respect to topology, there is a dichotomy. When the parameter $s$ is such that the critical orbit of $f_s$ is not dense, $\mathcal{GR}$ contains a dense $G_\delta$ set. In contrast, when the critical orbit of $f_s$ is dense, the complement of $\mathcal{GR}$ contains a dense $G_\delta$ set.

math.DS↗

arXiv2017

Intersections of multicurves from Dynnikov coordinates

We present an algorithm for calculating the geometric intersection number of two multicurves on the $n$-punctured disk, taking as input their Dynnikov coordinates. The algorithm has complexity $O(m^2n^4)$, where $m$ is the sum of the absolute values of the Dynnikov coordinates of the two multicurves. The main ingredient is an algorithm due to Cumplido for relaxing a multicurve.

math.GT↗

arXiv2017

Natural extensions of unimodal maps: virtual sphere homeomorphisms and prime ends of basin boundaries

Let $\{f_t\colon I\to I\}$ be a family of unimodal maps with topological entropies $h(f_t)>\frac12\log 2$, and ${\widehat{f}}_t\colon{\widehat{I}}_t\to{\widehat{I}}_t$ be their natural extensions, where ${\widehat{I}}_t=\varprojlim(I,f_t)$. Subject to some regularity conditions, which are satisfied by tent maps and quadratic maps, we give a complete description of the prime ends of the Barge-Martin embeddings of ${\widehat{I}}_t$ into the sphere. We also construct a family $\{\chi_t\colon S^2\to S^2\}$ of sphere homeomorphisms with the property that each $\chi_t$ is a factor of ${\widehat{f}}_t$, by a semi-conjugacy for which all fibers except one contain at most three points, and for which the exceptional fiber carries no topological entropy: that is, unimodal natural extensions are virtually sphere homeomorphisms. In the case where $\{f_t\}$ is the tent family, we show that $\chi_t$ is a generalized pseudo-Anosov map for the dense set of parameters for which $f_t$ is post-critically finite, so that $\{\chi_{t}\}$ is the completion of the unimodal generalized pseudo-Anosov family introduced in [21].

math.DS↗

arXiv2017

Itineraries for Inverse Limits of Tent Maps: a Backward View

Previously published admissibility conditions for an element of $\{0,1\}^{\mathbb{Z}}$ to be the itinerary of a point of the inverse limit of a tent map are expressed in terms of forward orbits. We give necessary and sufficient conditions in terms of backward orbits, which is more natural for inverse limits. These backward admissibility conditions are not symmetric versions of the forward ones: in particular, the maximum backward itinerary which can be realised by a tent map mode locks on intervals of kneading sequences.

math.DS↗

arXiv2015

Counting components of an integral lamination

We present an efficient algorithm for calculating the number of components of an integral lamination on an $n$-punctured disk, given its Dynnikov coordinates. The algorithm requires $O(n^2M)$ arithmetic operations, where $M$ is the sum of the absolute values of the Dynnikov coordinates.

math.GT↗

arXiv2015

Braid Equivalence in the H\'enon Family I

We give two general constructions of braid equivalences which exist between certain deformations of the 2-branched Horsehoe map. We then give numerical evidence suggesting that these constructions of braid equivalences are always realised in the H\'enon family.

math.DS↗

arXiv2014

New Rotation Sets in a Family of Torus Homeomorphisms

We construct a family $\{\Phi_t\}_{t\in[0,1]}$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $\rho(\Phi_t)$ can be described explicitly. We analyze the bifurcations and typical behavior of rotation sets in the family, providing insight into the general questions of toral rotation set bifurcations and prevalence. We show that there is a full measure subset of $[0,1]$, consisting of infinitely many mutually disjoint non-trivial closed intervals, on each of which the rotation set mode locks to a constant polygon with rational vertices; that the generic rotation set in the Hausdorff topology has infinitely many extreme points, accumulating on a single totally irrational extreme point at which there is a unique supporting line; and that, although $\rho(t)$ varies continuously with $t$, the set of extreme points of $\rho(t)$ does not. The family also provides examples of rotation sets for which an extreme point is not represented by any minimal invariant set, or by any directional ergodic measure.

math.DS↗

arXiv2013

On digit frequencies in {\beta}-expansions

We study the sets DF({\beta}) of digit frequencies of {\beta}-expansions of numbers in [0,1]. We show that DF({\beta}) is a compact convex set with countably many extreme points which varies continuously with {\beta}; that there is a full measure collection of non-trivial closed intervals on each of which DF({\beta}) mode locks to a constant polytope with rational vertices; and that the generic digit frequency set has infinitely many extreme points, accumulating on a single non-rational extreme point whose components are rationally independent.

math.DS↗

arXiv2013

Symbol ratio minimax sequences in the lexicographic order

Consider the space of sequences of k letters ordered lexicographically. We study the set M({\alpha}) of all maximal sequences for which the asymptotic proportions {\alpha} of the letters are prescribed, where a sequence is said to be maximal if it is at least as great as all of its tails. The infimum of M({\alpha}) is called the {\alpha}-infimax sequence, or the {\alpha}-minimax sequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax is not a minimax if and only if it is the {\alpha}-infimax for every {\alpha} in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms.

math.DS↗

arXiv2012

Inverse limits as attractors in parameterized families

We show how a parameterized family of maps of the spine of a manifold can be used to construct a family of homeomorphisms of the ambient manifold which have the inverse limits of the spine maps as global attractors. We describe applications to unimodal families of interval maps, to rotation sets, and to the standard family of circle maps.

math.DS↗

arXiv2011

Riemann surfaces out of paper

Let S be a surface obtained from a plane polygon by identifying infinitely many pairs of segments along its boundary. A condition is given under which the complex structure in the interior of the polygon extends uniquely across the quotient of its boundary to make S into a closed Riemann surface. When this condition holds, a modulus of continuity is obtained for a uniformizing map on S.

math.CV↗

arXiv2010

Paper folding, Riemann surfaces, and convergence of pseudo-Anosov sequences

A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees that the Euclidean structure on the polygons induces a unique conformal structure on the quotient surface, making it into a closed Riemann surface. In this case, a modulus of continuity for uniformizing coordinates is found which depends only on the geometry of the polygons and on the identifications. An application is presented in which a uniform modulus of continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it possible to prove that they converge to a Teichm\"uller mapping on the Riemann sphere.

math.DS↗

arXiv2008

Topology of Chaotic Mixing Patterns

A stirring device consisting of a periodic motion of rods induces a mapping of the fluid domain to itself, which can be regarded as a homeomorphism of a punctured surface. Having the rods undergo a topologically-complex motion guarantees at least a minimum amount of stretching of material lines, which is important for chaotic mixing. We use topological considerations to describe the nature of the injection of unmixed material into a central mixing region, which takes place at injection cusps. A topological index formula allow us to predict the possible types of unstable foliations that can arise for a fixed number of rods.

nlin.CD↗

arXiv2008

On the topological entropy of families of braids

A method for computing the topological entropy of each braid in an infinite family, making use of Dynnikov's coordinates on the boundary of Teichmüller space, is described. The method is illustrated on two two-parameter families of braids.

math.DS↗