Searcharxiv⌕ Search

arXiv subjects

Danny Calegari

Publications and source records attributed to Danny Calegari.

At least 19 recordsLinked to original sources
arXiv2026

Laminations and External Angles for Similarity Pairs

A {\em similarity pair} is the dynamical system in $\mathbb{C}$ generated by two maps $f:z \to sz-1$ and $g:z \to sz+1$ for $|s|<1$. Associated to the dynamical system is an attractor $\Lambda$. The Barnsley--Harrington Mandelbrot set $\mathcal{M}$ is the set of $s\in \mathbb{D}$ for which $\Lambda$ is connected. Let $K$ denote the filled set of a connected $\Lambda$. For $s\in \partial \mathcal{M}$ we show that the (partially defined) action of the semigroup on $\partial K$ is topologically conjugate to a (discontinuous) piecewise linear action of constant slope. Conditional on a conjecture (satisfied for `most' $s\in \partial \mathcal{M}$) we give a necessary and sufficient condition in terms of the dynamics on $\partial K$ for $K$ to contain cut points, and we describe the set of all such cut points in terms of infinite walks in a directed graph $\textrm{IG}$ obtained by an explicit recursive algorithm. The structure of the `dynamical cut point set' for a 2-dimensional family of piecewise linear actions (containing those coming from $s\in \partial \mathcal{M}$) recovers and generalizes the Douady--Hubbard--Thurston quadratic minor lamination for the abstract Mandelbrot set.

math.DS↗
arXiv2026

Quasimorphisms and Pseudo-Anosov flows

We describe two connections between the theory of quasimorphisms and pseudo-Anosov flows without perfect fits on closed hyperbolic 3-manifolds. First we show that for every such flow $X$, there are quasimorphisms whose coarse restriction to each flowline of $\tilde{X}$ (the lifted flow in the universal cover) are uniform quasi-isometries to $\mathbb{R}$ -- such quasimorphisms are said to be *adapted* to $X$; and that the space of quasimorphisms $Q_X$ adapted to $X$ is an open convex cone in the space of all quasimorphisms on $\pi_1(M)$. Second, we obtain upper bounds on the exponential growth rate of closed orbits in such flows, both in the hyperbolic metric and in a word metric; quasimorphisms play a key role in obtaining the estimates in the second case.

math.DS↗
arXiv2026

CaTherine wheels

A CaTherine wheel is a surjective continuous map $f:S^1 \to S^2$ such that for every closed interval $I\subset S^1$ the image $f(I)$ is homeomorphic to a disk, and $f(\partial I)$ is contained in the boundary of this disk. CaTherine wheels arise in many areas of low-dimensional geometry and topology, including conformal dynamics (expanding Thurston maps, expanding origamis), probability theory (whole plane ${\rm SLE}_\kappa$ for $\kappa \ge 8$, LQG metric trees) and elsewhere. We develop their theory in generality, and explain how CaTherine wheels and their associated structures can serve as a dictionary between these various fields. Our most substantial applications are to the theory of hyperbolic 3-manifolds. If $M$ is a closed hyperbolic 3-manifold and $G=\pi_1(M)$, we show that there is a canonical bijection between four kinds of structures associated to $M$: 1. orbit-equivalence classes of pseudo-Anosov flows on $M$ without perfect fits; 2. $G$-equivariant CaTherine wheels up to conjugacy; 3. minimal $G$-zippers; and 4. connected components of the space of uniform quasimorphisms on $G$. This generalizes and amplifies the theory of fiberings of hyperbolic 3-manifolds over the circle and the Thurston norm.

math.GT↗
arXiv2026

CaTherine wheels from trees and Liouville quantum gravity

A CaTherine wheel is a space-filling curve $f : S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\partial J)$ is contained in $\partial f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at $\infty$ for the $\gamma$-Liouville quantum gravity (LQG) metric, for $\gamma \in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree.

math.PR↗
arXiv2025

Triangulating PL functions and the existence of efficient ReLU DNNs

We show that every piecewise linear function $f:R^d \to R$ with compact support a polyhedron $P$ has a representation as a sum of so-called `simplex functions'. Such representations arise from degree 1 triangulations of the relative homology class (in $R^{d+1}$) bounded by $P$ and the graph of $f$, and give a short elementary proof of the existence of efficient universal ReLU neural networks that simultaneously compute all such functions $f$ of bounded complexity.

cs.LG↗
arXiv2024

Zippers

If $M$ is a hyperbolic 3-manifold fibering over the circle, the fundamental group of $M$ acts faithfully by homeomorphisms on a circle (the circle at infinity of the universal cover of the fiber), preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures (e.g. taut foliations, quasigeodesic or pseudo-Anosov flows) are known to give rise to universal circles -- a circle with a faithful $\pi_1(M)$ action preserving a pair of invariant laminations -- and these universal circles play a key role in relating the dynamical structure to the geometry of $M$. In this paper we introduce the idea of zippers, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers (and their associated universal circles) may be constructed directly from uniform quasimorphisms or from uniform left orders.

math.GT↗
arXiv2022

Wiggle Island

A wiggle is an embedded curve in the plane that is the attractor of an iterated function system associated to a complex parameter z. We show the space of wiggles is disconnected -- i.e. there is a wiggle island.

math.DS↗
arXiv2022

Surgery sequences and self-similarity of the Mandelbrot set

We introduce an analog in the context of rational maps of the idea of hyperbolic Dehn surgery from the theory of Kleinian groups. A surgery sequence is a sequence of postcritically finite maps limiting (in a precise manner) to a postcritically finite map with at least one strictly preperiodic critical orbit. As an application of this idea we give a new and elementary proof of Tan Lei's theorem on the asymptotic self-similarity of Julia Sets and the Mandelbrot Set at Misiurewicz points.

math.DS↗
arXiv2022

Normal subgroups of big mapping class groups

Let S be a surface and let Mod(S,K) be the mapping class group of S permuting a Cantor subset K of S. We prove two structure theorems for normal subgroups of Mod(S,K). (Purity:) if S has finite type, every normal subgroup of Mod(S,K) either contains the kernel of the forgetful map to the mapping class group of S, or it is `pure', i.e. it fixes the Cantor set pointwise. (Inertia:) for any n element subset Q of the Cantor set, there is a forgetful map from the pure subgroup PMod(S,K) of Mod(S,K) to the mapping class group of (S,Q) fixing Q pointwise. If N is a normal subgroup of Mod(S,K) contained in PMod(S,K), its image N_Q is likewise normal. We characterize exactly which finite-type normal subgroups N_Q arise this way. Several applications and numerous examples are also given.

math.GT↗
arXiv2022

Sausages and Butcher Paper

For each $d>1$ the shift locus of degree $d$, denoted ${\mathcal S}_d$, is the space of normalized degree $d$ polynomials in one complex variable for which every critical point is in the attracting basin of infinity under iteration. It is a complex analytic manifold of complex dimension $d-1$. We are able to give an explicit description of ${\mathcal S}_d$ as a complex of spaces over a contractible $\tilde{A}_{d-2}$ building, and to describe the pieces in two quite different ways: 1. (combinatorial): in terms of dynamical extended laminations; or 2. (algebraic): in terms of certain explicit `discriminant-like' affine algebraic varieties. From this structure one may deduce numerous facts, including that ${\mathcal S}_d$ has the homotopy type of a CW complex of real dimension $d-1$; and that ${\mathcal S}_3$ and ${\mathcal S}_4$ are $K(π,1)$s. The method of proof is rather interesting in its own right. In fact, along the way we discover a new class of complex surfaces (they are complements of certain singular curves in ${\mathbb C}^2$) which are homotopic to locally CAT$(0)$ complexes; in particular they are $K(π,1)$s.

math.DS↗
arXiv2021

Sausages

The shift locus is the space of normalized polynomials in one complex variable for which every critical point is in the attracting basin of infinity. The method of sausages gives a (canonical) decomposition of the shift locus in each degree into (countably many) codimension 0 submanifolds, each of which is homeomorphic to a complex algebraic variety. In this paper we explain the method of sausages, and some of its consequences. This is an expository paper.

math.DS↗
arXiv2021

Combinatorics of the Tautological Lamination

The Tautological Lamination arises in holomorphic dynamics as a combinatorial model for the geometry of 1-dimensional slices of the Shift Locus. In each degree $q$ the tautological lamination defines an iterated sequence of partitions of $1$ (one for each integer $n$) into numbers of the form $2^m q^{-n}$. Denote by $N_q(n,m)$ the number of times $2^mq^{-n}$ arises in the $n$th partition. We prove a recursion formula for $N_q(n,0)$, and a gap theorem: $N_q(n,n)=1$ and $N_q(n,m)=0$ for $\lfloor n/2 \rfloor < m < n$.

math.DS↗
arXiv2020

Nielsen realization for infinite-type surfaces

Given a finite subgroup G of the mapping class group of a surface S, the Nielsen realization problem asks whether G can be realized as a finite group of homeomorphisms of S. In 1983, Kerckhoff showed that for S a finite-type surface, any finite subgroup G may be realized as a group of isometries of some hyperbolic metric on S. We extend Kerckhoff's result to orientable, infinite-type surfaces. As applications, we classify torsion elements in the mapping class group of the plane minus a Cantor set, and also show that topological groups containing sequences of torsion elements limiting to the identity do not embed continuously into the mapping class group of S. Finally, we show that compact subgroups of the mapping class group of S are finite, and locally compact subgroups are discrete.

math.GT↗
arXiv2020

Taut foliations leafwise branch cover S^2

A co-oriented foliation F of an oriented 3-manifold M is taut if and only if there is a map from M to the 2-sphere whose restriction to every leaf is a branched cover.

math.GT↗
arXiv2020

Counting minimal surfaces in negatively curved 3-manifolds

We introduced an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature less than or equal -1, by the hyperbolic metric.

math.DG↗
arXiv2018

Extreme points in limit sets

Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.

math.DS↗
arXiv2017

3-manifolds everywhere

A random group contains many subgroups which are isomorphic to the fundamental group of a compact hyperbolic 3-manifold with totally geodesic boundary. These subgroups can be taken to be quasi-isometrically embedded. This is true both in the few relators model, and the density model of random groups (at any density less than a half).

math.GR↗