An official website of the United States government
The Haefliger–Thurston conjecture predicts that Haefliger's classifying space for$$C^r$$-foliations of codimension$$n$$whose normal bundles are trivial is$$2n$$-connected. In this paper, we confirm this conjecture for piecewise linear (PL) foliations of codimension$$2$$. Using this, we use a version of the Mather–Thurston theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer the question of Epstein in dimension$$2$$and prove the simplicity of the identity component of PL surface homeomorphisms.
more »
« less
This content will become publicly available on August 1, 2026
Lavaurs algorithm for cubic symmetric polynomials
Abstract As discovered by W. Thurston, the action of a complex one-variable polynomial on its Julia set can be modeled by a geodesic lamination in the disk, provided that the Julia set is connected. It also turned out that the parameter space of such dynamical laminations of degree two gives a model for the bifurcation locus in the space of quadratic polynomials. This model is itself a geodesic lamination, the so calledquadratic minor laminationof Thurston. In the same spirit, we consider the space of allcubic symmetric polynomials$$f_\unicode{x3bb} (z)=z^3+\unicode{x3bb} ^2 z$$in three articles. In the first one, we construct thecubic symmetric comajor laminationtogether with the corresponding quotient space of the unit circle. As is verified in the third paper, this yields a monotone model of thecubic symmetric connectedness locus, that is, the space of all cubic symmetric polynomials with connected Julia sets. In the present paper, the second in the series, we develop an algorithm for generating the cubic symmetric comajor lamination analogous to the Lavaurs algorithm for constructing the quadratic minor lamination.
more »
« less
- Award ID(s):
- 2349942
- PAR ID:
- 10613024
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 45
- Issue:
- 8
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 2314 to 2340
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract Every Thurston map$$f\colon S^2\rightarrow S^2$$on a$$2$$-sphere$$S^2$$induces a pull-back operation on Jordan curves$$\alpha \subset S^2\smallsetminus {P_f}$$, where$${P_f}$$is the postcritical set off. Here the isotopy class$$[f^{-1}(\alpha )]$$(relative to$${P_f}$$) only depends on the isotopy class$$[\alpha ]$$. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the mapfcan be seen as a fixed point of the pull-back operation. We show that if a Thurston mapfwith a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying$$2$$-sphere and construct a new Thurston map$$\widehat f$$for which this obstruction is eliminated. We prove that no other obstruction arises and so$$\widehat f$$is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.more » « less
-
Abstract Let$$\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$$be integral linear combinations of elementary symmetric polynomials with$$\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$$, where$$1\le k_1<\cdots . Subject to the condition$$k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$$, we show that there is a paucity of nondiagonal solutions to the Diophantine system$$\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$$.more » « less
-
Abstract We give an explicit raising operator formula for the modified Macdonald polynomials$$\tilde {H}_{\mu }(X;q,t)$$, which follows from our recent formula for$$\nabla $$on an LLT polynomial and the Haglund-Haiman-Loehr formula expressing modified Macdonald polynomials as sums of LLT polynomials. Our method just as easily yields a formula for a family of symmetric functions$$\tilde {H}^{1,n}(X;q,t)$$that we call$$1,n$$-Macdonald polynomials, which reduce to a scalar multiple of$$\tilde {H}_{\mu }(X;q,t)$$when$$n=1$$. We conjecture that the coefficients of$$1,n$$-Macdonald polynomials in terms of Schur functions belong to$${\mathbb N}[q,t]$$, generalizing Macdonald positivity.more » « less
-
Abstract For$$g\ge 2$$and$$n\ge 0$$, let$$\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$$denote the complex moduli stack ofn-marked smooth hyperelliptic curves of genusg. A normal crossings compactification of this space is provided by the theory of pointed admissible$$\mathbb {Z}/2\mathbb {Z}$$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of$$\mathcal {H}_{g, n}$$. Using this graph complex, we give a sum-over-graphs formula for the$$S_n$$-equivariant weight zero compactly supported Euler characteristic of$$\mathcal {H}_{g, n}$$. This formula allows for the computer-aided calculation, for each$$g\le 7$$, of the generating function$$\mathsf {h}_g$$for these equivariant Euler characteristics for alln. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissibleG-covers of genus zero curves, whenGis abelian, as a symmetric$$\Delta $$-complex. We use these complexes to generalize our formula for$$\mathsf {h}_g$$to moduli spaces ofn-pointed smooth abelian covers of genus zero curves.more » « less
