arXiv subjects

Philip Boyland

Publications and source records attributed to Philip Boyland.

At least 19 recordsLinked to original sources

arXiv2025

Pinched Arnol'd tongues for Families of circle maps

The family of circle maps \begin{equation*} f_{b, ω} (x) = x + ω+ b\, ϕ(x) \end{equation*} is used as a simple model for a periodically forced oscillator. The parameter $ω$ represents the unforced frequency, $b$ the coupling, and $ϕ$ the forcing. When $ϕ= \frac{1}{2 π} \sin(2 πx)$ this is the classical Arnol'd standard family. Such families are often studied in the $(ω,b)$-plane via the so-called tongues $T_β$ consisting of all $(ω,b)$ such that $f_{b, ω}$ has rotation number $β$. The interior of the rational tongues $T_{p/q}$ represent the system mode-locked into a $p/q$-periodic response. Campbell, Galeeva, Tresser, and Uherka proved that when the forcing is a PL map with $k=2$ breakpoints, all $T_{p/q}$ pinch down to a width of a single point at multple values when $q$ large enough. In contrast, we prove that it generic amongst PL forcings with a given $k\geq 3$ breakpoints that there is no such pinching of any of the rational tongues. We also prove that the absence of pinching is generic for Lipschitz and $C^r$ ($r>0$) forcing.

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arXiv2025

Unimodal Measurable Pseudo-Anosov Maps

We exhibit a continuously varying family $F_λ$ of homeomorphisms of the sphere $S^2$, for which each $F_λ$ 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 $F_λ$ is semi-conjugate to the inverse limit of the core tent map with slope $λ$: it is topologically transitive, ergodic with respect to a background Oxtoby-Ulam measure, has dense periodic points, and has topological entropy $h(F_λ) = \log λ$ (so that no two $F_λ$ are topologically conjugate). For a full measure, dense $G_δ$ set of parameters, $F_λ$ is a measurable pseudo-Anosov map but not a generalized pseudo-Anosov map, and its turbulations are nowhere locally regular.

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arXiv2024

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.

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arXiv2024

Dynamical displacements, persistence and semiconjugacies

This survey gives a unified treatment of topics from Abelian and non-Abelian Nielsen Theory integrated with the semiconjugacy theorems of Franks and Handel. The main focus is to develop an analog of the rotation set that is valid when the dynamics are not isotopic to the identity and to connect this theory to the dynamical persistence under homotopy/isotopy intrinsic in the theorems of Franks and Handel. For this dynamical persistence, expansion/hyperbolicity at some scale is essential.

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arXiv2022

On the abundance of $k$-fold semi-monotone minimal sets in bimodal circle maps

Inspired by a twist maps theorem of Mather we study recurrent invariant sets that are ordered like rigid rotation under the action of the lift of a bimodal circle map $g$ to the $k$-fold cover. For each irrational in the interior of the rotation set the collection of the $k$-fold ordered semi-Denjoy minimal sets with that rotation number contains a $(k-1)$-dimensional ball in the weak topology on their unique invariant measures. We also describe completely their periodic orbit analogs for rational rotation numbers. The main tool is a generalization of a construction of Hedlund and Morse which generates the symbolic analogs of these $k$-fold well-ordered invariant sets.

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arXiv2020

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 $G_δ$ set. In contrast, when the critical orbit of $f_s$ is dense, the complement of $\mathcal{GR}$ contains a dense $G_δ$ set.

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arXiv2020

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 $\{χ_t\colon S^2\to S^2\}$ of sphere homeomorphisms with the property that each $χ_t$ 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 $χ_t$ is a generalized pseudo-Anosov map for the dense set of parameters for which $f_t$ is post-critically finite, so that $\{χ_{t}\}$ is the completion of the unimodal generalized pseudo-Anosov family introduced in [21].

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arXiv2019

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 $Φ_t\colon D^2\to D^2$, having transitive global attractors $Λ_t$ on which $Φ_t$ 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 $Φ_t$-invariant measures $ν_t$, supported on the attractors $Λ_t$. We show that this family $ν_t$ varies weakly continuously, and that the measures $ν_t$ are physical with respect to a weakly continuously varying family of background Oxtoby-Ulam measures $ρ_t$. Similar results are obtained for the family $χ_t\colon S^2\to S^2$ of transitive sphere homeomorphisms, constructed in [17] as factors of the natural extensions of $f_t$.

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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.

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arXiv2017

Geometric representation of the infimax S-adic family

We construct geometric realizations for the infimax family of substitutions by generalizing the Rauzy-Canterini-Siegel method for a single substitution to the S-adic case. The composition of each countably infinite subcollection of substitutions from the family has an asymptotic fixed sequence whose shift orbit closure is an infimax minimal set $Δ^+$. The subcollection of substitutions also generates an infinite Bratteli-Vershik diagram with prefix-suffix labeled edges. Paths in the diagram give the Dumont-Thomas expansion of sequences in $Δ^+$ which in turn gives a projection onto the asymptotic stable direction of the infinite product of the Abelianization matrices. The projections of all sequences from $Δ^+$ is the generalized Rauzy fractal which has subpieces corresponding to the images of symbolic cylinder sets. The intervals containing these subpieces are shown to be disjoint except at endpoints, and thus the induced map derived from the symbolic shift translates them. Therefore the process yields an Interval Translation Map, and the Rauzy fractal is proved to be its attractor.

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arXiv2015

New Rotation Sets in a Family of Torus Homeomorphisms

We construct a family $\{Φ_t\}_{t\in[0,1]}$ of homeomorphisms of the two-torus isotopic to the identity, for which all of the rotation sets $ρ(Φ_t)$ 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 $ρ(t)$ varies continuously with $t$, the set of extreme points of $ρ(t)$ 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.

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arXiv2014

On digit frequencies in β-expansions

We study the sets DF(β) of digit frequencies of β-expansions of numbers in [0,1]. We show that DF(β) is a compact convex set with countably many extreme points which varies continuously with β; that there is a full measure collection of non-trivial closed intervals on each of which DF(β) 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.

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arXiv2013

Symbol ratio minimax sequences in the lexicographic order

Consider the space of sequences of k letters ordered lexicographically. We study the set M(α) of all maximal sequences for which the asymptotic proportions α 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(α) is called the α-infimax sequence, or the α-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 α-infimax for every α in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms.

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arXiv2012

Exponential growth in two-dimensional topological fluid dynamics

This paper describes topological kinematics associated with the stirring by rods of a two-dimensional fluid. The main tool is the Thurston-Nielsen (TN) theory which implies that depending on the stirring protocol the essential topological length of material lines grows either exponentially or linearly. We give an application to the growth of the gradient of a passively advected scalar, the Helmholtz-Kelvin Theorem then yields applications to Euler flows. The main theorem shows that there are periodic stirring protocols for which generic initial vorticity yields a solution to Euler's equations which is not periodic and further, the $L^\infty$ and $L^1$-norms of the gradient of its vorticity grow exponentially in time.

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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.

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arXiv2011

The entropy efficiency of point-push mapping classes on the punctured disk

We study the maximal entropy per unit generator of push-point mapping classes on the punctured disk. Our work is motivated by fluid mixing by rods in a planar domain. If a single rod moves among N-fixed obstacles, the resulting fluid diffeomorphism is in the push-point mapping class associated with the loop in π_1(D^2 - {N points}) traversed by the single stirrer. The collection of motions in each of which the stirrer goes around a single obstacle generate the group of push-point mapping classes, and the entropy efficiency with respect to these generators gives a topological measure of the mixing per unit energy expenditure of the mapping class. We give lower and upper bounds for Eff(N), the maximal efficiency in the presence of N obstacles, and prove that Eff(N) -> log(3) as N -> \infty. For the lower bound we compute the entropy efficiency of a specific push-point protocol, HSP_N, which we conjecture achieves the maximum. The entropy computation uses the action on chains in a \Z-covering space of the punctured disk which is designed for push-point protocols. For the upper bound we estimate the exponential growth rate of the action of the push-point mapping classes on the fundamental group of the punctured disk using a collection of incidence matrices and then computing the generalized spectral radius of the collection.

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arXiv2010

On eigen-structures for pseudoAnosov maps

We investigate various structures associated with the hyperbolic Markov and homological spectra of a pseudoAnosov map $ϕ$ on a surface. Each unstable eigenvalue of the action of $ϕ$ on first cohomolgy yields an eigen-cocycle that is transverse and holonomy invariant to the stable foliation $\mathcal{F}^s$ of $ϕ$. Each unstable eigenvalue $μ$ of a Markov transition matrix for $ϕ$ yields a holonomy invariant additive function $G$ on transverse arcs to $\cF^s$ with $ϕ^* G = μG$. Except when $μ$ is the dilation of $ϕ$, these transverse arc functions do not yield measures, but rather holonomy invariant eigen-distributions which are dual to Hölder functions. Stable homological and Markov eigenvalues yield analogous transverse structures to the unstable foliation of $ϕ$. The main tool for working with the homological spectrum is the Franks-Shub Theorem which holds for a general manifold and map. For the Markov spectrum we use the correspondence of the leaf space of stable foliation with a one-sided subshift of finite type. This identification allows the symbolic analog of a transverse arc function to be defined, analyzed, and applied.

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arXiv2008

Transitivity of Surface Dynamics Lifted to Abelian Covers

A homeomorphism f of a manifold M is called H_1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover \tM. Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H_1(M). For a rel pseudo-Anosov map ϕof a compact surface M we show that the following are equivalent: (a) ϕis H_1-transitive, (b) the action of ϕon H_1(M) has spectral radius one, and (c) the lifts of the invariant foliations of ϕto \tM have dense leaves. The proof relies on a characterization of transitivity for twisted \Z^d-extensions of a transitive subshift of finite type.

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