Tootfinder

This https://arxiv.org/abs/2310.00778 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Minimal diffeomorphisms with $L^1$ Hopf differentials
We prove that for any two Riemannian metrics $σ_1, σ_2$ on the unit disk, a homeomorphism $\partial\mathbb{D}\to\partial\mathbb{D}$ extends to at most one quasiconformal minimal diffeomorphism $(\mathbb{D},σ_1)\to (\mathbb{D},σ_2)$ with $L^1$ Hopf differential. For minimal Lagrangian diffeomorphisms between hyperbolic disks, the result is known, but this is the first proof that does not use anti-de Sitter geometry. We show that the result fails without the $L^1$ assumption in variable curva…

Can points of bounded orbits surround points of unbounded orbits ?
Jiehua Mai, Enhui Shi, Kesong Yan, Fanping Zeng
https://arxiv.org/abs/2404.13642 https:/…

Can points of bounded orbits surround points of unbounded orbits ?
We show a somewhat surprising result: if $E$ is a disk in the plane $\mathbb R^2$, then there is a homeomorphism $h:\mathbb R^2\rightarrow\mathbb R^2$ such that, for every $x\in\partial E$, the orbit $O(x, h)$ is bounded, but for every $y\in {\rm Int}(E)$, the orbit $O(y, h)$ is doubly divergent. To prove this, we define a class of homeomorphisms on the square $[-1, 1]^2$, called normally rising homeomorphisms, and show that a normally rising homeomorphism can have very complex $ω$-limit sets …

This https://arxiv.org/abs/2311.06857 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Fusion-equivariant stability conditions and Morita duality
Given a triangulated category $D$ with an action of a fusion category $C$, we study the moduli space $Stab_{C}(D)$ of fusion-equivariant Bridgeland stability conditions on $D$. The main theorem is that the fusion-equivariant stability conditions form a closed, complex submanifold of the moduli space of stability conditions on $D$. As an application of this framework, we generalise a result of Macrì--Mehrotra--Stellari by establishing a homeomorphism between the space of $G$-invariant stability…

Computing shortest closed curves on non-orientable surfaces
Denys Bulavka, \'Eric Colin de Verdi\`ere, Niloufar Fuladi
https://arxiv.org/abs/2403.11749

Computing shortest closed curves on non-orientable surfaces
We initiate the study of computing shortest non-separating simple closed curves with some given topological properties on non-orientable surfaces. While, for orientable surfaces, any two non-separating simple closed curves are related by a self-homeomorphism of the surface, and computing shortest such curves has been vastly studied, for non-orientable ones the classification of non-separating simple closed curves up to ambient homeomorphism is subtler, depending on whether the curve is one-side…

This https://arxiv.org/abs/2305.10908 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Exotic 4-manifolds with signature zero
We produce infinitely many distinct irreducible smooth 4-manifolds homeomorphic to #(2m+1)(CP^2 # -CP^2) and #(2n+1)(S^2 x S^2), respectively, for each m>3 and n>4. These provide the smallest exotic closed simply connected 4-manifolds with signature zero known to date, and in each one of these homeomorphism classes, we get minimal symplectic 4-manifolds. Our novel exotic 4-manifolds are derived from fairly special small Lefschetz fibrations we build via positive factorizations in the mapping cl…

On the topological classification of complex plane curve singularities
Alberto Fern\'andez-Hern\'andez
https://arxiv.org/abs/2402.13941 https://

On the topological classification of complex plane curve singularities
This final degree project is devoted to study the topological classification of complex plane curves. These are subsets of $\mathbb{C}^2$ that can be described by an equation $f(x,y)=0$. Loosely speaking, curves are said to be equivalent in a topological sense whenever they are ambient homeomorphic, i.e., there exists an orientation-preserving homeomorphism of the ambient space carrying one curve to the other. The project's aim is to develop operative and clear conditions to determine whether t…

This https://arxiv.org/abs/1909.03641 has been replaced.
link: https://scholar.google.com/scholar?q=a

Definable (co)homology, pro-torus rigidity, and (co)homological classification
We show that the classical homology theory of Steenrod may be enriched with descriptive set-theoretic information. We prove that the resulting definable homology theory provides a strictly finer invariant than Steenrod homology for compact metrizable spaces up to homotopy. In particular, we show that pro-tori are completely classified up to homeomorphism by their definable homology. This is in contrast with the fact that, for example, there exist uncountably many pairwise non-homeomorphic solen…

This https://arxiv.org/abs/2403.08667 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Surfaces and other Peano Continua with no Generic Chains
The space of chains on a compact connected space encodes all the different ways of continuously growing out of a point until exhausting the space. A chain is generic if its orbit under the action of the underlying homeomorphism group is comeager: a space X has a generic chain if there is essentially just one type of chain on X. Gutman, Tsankov and Zucker proved that compact manifolds of dimension at least 3 do not have a generic chain. We extend and generalize their result, covering a large cla…

Homeomorphism groups of telescoping 2-manifolds are strongly distorted
Nicholas G. Vlamis
https://arxiv.org/abs/2403.03887 https://ar…

Homeomorphism groups of telescoping 2-manifolds are strongly distorted
Building on the work of Mann and Rafi, we introduce an expanded definition of a telescoping 2-manifold and proceed to study the homeomorphism group of a telescoping 2-manifold. Our main result shows that it is strongly distorted. We then give a simple description of its commutator subgroup, which is index one, two, or four depending on the topology of the manifold. Moreover, we show its commutator subgroup is uniformly perfect with commutator width at most two, and we give a family of uniform n…

Cohomologically smooth morphisms for \'etale $\mathbb F_p$-sheaves in characteristic $p$
Felix Lotter
https://arxiv.org/abs/2403.11149 https://<…

Cohomologically smooth morphisms for étale $\mathbb F_p$-sheaves in characteristic $p$
We scrutinise the notions of cohomologically smooth morphisms and smooth objects for the six functor formalism of étale $\mathbb F_p$-sheaves on schemes in characteristic $p$. We show that only cohomologically étale morphisms are cohomologically smooth in this setting. This is complemented by a characterisation of cohomologically étale morphisms in arbitrary characteristic. In fact, we prove that such a morphism is already étale up to universal homeomorphism.

This https://arxiv.org/abs/2201.02598 has been replaced.
link: https://scholar.google.com/scholar?q=a

Completeness of derived interleaving distances and sheaf quantization of non-smooth objects
We develop sheaf-theoretic methods to deal with non-smooth objects in symplectic geometry. We show the completeness of a derived category of sheaves with respect to the interleaving distance and construct a sheaf quantization of a Hamiltonian homeomorphism. We also develop Lusternik--Schnirelmann theory in the microlocal theory of sheaves. With these new sheaf-theoretic methods, we prove an Arnold-type theorem for the image of a compact exact Lagrangian submanifold under a Hamiltonian homeomorp…

This https://arxiv.org/abs/2306.08619 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Homeomorphism groups of self-similar 2-manifolds
The class of self-similar 2-manifolds consists of manifolds exhibiting a type of homogeneity akin to the 2-sphere and the Cantor set, and includes both the 2-sphere and the 2-sphere with a Cantor set removed. This chapter aims to provide a narrative thread between recent results on the structure of homeomorphism groups/mapping class groups of self-similar 2-manifolds, and also connections to classical structural results on the homeomorphism group of the 2-sphere and the Cantor set. In order to …

Irregular sets for a class of skew product transformations
Nian Liu, Xue Liu
https://arxiv.org/abs/2404.12562 https://arxiv.org/pdf/2…

Irregular sets for a class of skew product transformations
In this paper, we study the irregular set of any continuous observable for a class of skew product transformations, which is driven by a uniquely ergodic homeomorphism system $(Ω,\mathbb{P},θ)$ and satisfies Anosov and toplogical mixing on fibers property. We prove that if the irregular set of any continuous observable is nonempty on a fiber, then the irregular set must be nonempty, residual, and carry full fiber topological entropy on $\mathbb{P}$-a.e. fibers. The orbit-gluing technique prov…

This https://arxiv.org/abs/2306.08619 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Homeomorphism groups of self-similar 2-manifolds
The class of self-similar 2-manifolds consists of manifolds exhibiting a type of homogeneity akin to the 2-sphere and the Cantor set, and includes both the 2-sphere and the 2-sphere with a Cantor set removed. This chapter aims to provide a narrative thread between recent results on the structure of homeomorphism groups/mapping class groups of self-similar 2-manifolds, and also connections to classical structural results on the homeomorphism group of the 2-sphere and the Cantor set. In order to …

Genericity of homeomorphisms with full mean Hausdorff dimension
Jeovanny de Jesus Muentes Acevedo
https://arxiv.org/abs/2402.14405 https://

Genericity of homeomorphisms with full mean Hausdorff dimension
It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs. Estimating the exact values of both the metric mean dimension and mean Hausdorff dimension for a homeomorphism is a challenging task. We need to establish a precise relationship between the sizes of the horseshoes and the number of appropriated legs to control bo…

This https://arxiv.org/abs/2307.01153 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Integral cohomology rings of weighted Grassmann orbifolds and rigidity properties
In this paper, we introduce `Plücker weight vector' and establish the definition of a weighted Grassmann orbifold $\mbox{Gr}_{\mathbf{b}}(k,n)$, corresponding to a Plücker weight vector `$\mathbf{b}$'. We achieve an explicit classification of weighted Grassmann orbifolds up to certain homeomorphism in terms of the Plücker weight vectors. We study the integral cohomology of $\mbox{Gr}_{\bf b}(k,n)$ and provide some sufficient conditions such that the integral cohomology of $\mbox{Gr}_{\bf b}(…

C^0 Lagrangian Monodromy
Noah Porcelli
https://arxiv.org/abs/2404.04591 https://arxiv.org/pdf/2404.04591

C^0 Lagrangian Monodromy
We prove that (under appropriate orientation assumptions), the action of a Hamiltonian homeomorphism ϕon the cohomology of a relatively exact Lagrangian fixed by ϕis the identity. This extends results of Hu-Lalonde-Leclercq and the author in the setting of Hamiltonian diffeomorphisms. We also prove a similar result regarding the action of ϕon relative cohomology.

This https://arxiv.org/abs/2107.09407 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Parabolic Mandelbrot Set
We solve the longstanding conjecture by Milnor (1993) concerning the connectedness locus $M_1$ of the family of quadratic rational maps tangent to the identity at $\infty$. We prove that this locus in homeomorphic to the Mandelbrot set $M$ and that the homeomorphism is unique, provided it identifies maps that are "hybridly" conjugate on their filled-in Julia set. Moreover this homeomorphism from $M$ to $M_1$ is nowhere Hölder on the boundary and so can not have even locally a quasi-conformal e…

This https://arxiv.org/abs/2206.15226 has been replaced.
link: https://scholar.google.com/scholar?q=a

Earthquake theorem for cluster algebras of finite type
We introduce a cluster algebraic generalization of Thurston's earthquake map for the cluster algebras of finite type, which we call the \emph{cluster earthquake map}. It is defined by gluing exponential maps, which is modeled after the earthquakes along ideal arcs. We prove an analogue of the earthquake theorem, which states that the cluster earthquake map gives a homeomorphism between the spaces of $\mathbb{R}^\mathrm{trop}$- and $\mathbb{R}_{>0}$-valued points of the cluster $\mathcal{X}$-var…

Orientation Preserving Homeomorphisms of the Plane having BP-Chain Recurrent Points
Jiehua Mai, Kesong Yan, Fanping Zeng
https://arxiv.org/abs/2404.10247 h…

Orientation Preserving Homeomorphisms of the Plane having BP-Chain Recurrent Points
More than a century ago, L. E. J. Brouwer proved a famous theorem, which says that any orientation preserving homeomorphism of the plane having a periodic point must have a fixed point. In recent years, there are still some authors giving various proofs of this fixed point theorem. In \cite{Fa}, Fathi showed that the condition``having a periodic point'' in this theorem can be weakened to ``having a non-wandering point''. In this paper, we first give a new proof of Brouwer's theorem, which is re…

This https://arxiv.org/abs/1209.2818 has been replaced.
link: https://scholar.google.com/scholar?q=a<…

A class of open surfaces with algorithmically solvable homeomorphism problem
We introduce a new class of possibly noncompact n-dimensional manifolds without boundary associated to finite data which we call topological automata. This class is large enough to contain many interesting examples of open 2-dimensional and 3-dimensional manifolds of interest to low-dimensional topologists. Our main result is that the homeomorphism problem in this class is decidable for n = 2.

This https://arxiv.org/abs/2303.11227 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The compact-open topology on the diffeomorphism or homeomorphism group of a smooth manifold without boundary is minimal in almost all dimensions
We show that for any connected smooth manifold $M$ of dimension different from $3$ the restriction of the compact-open topology to the diffeomorphism group of $M$ is minimal, i.e. the group does not admit a strictly coarser Hausdorff group topology. This implies the minimality of the compact-open topology on the homeomorphism group of $M$ in all dimensions different from $3$ and $4$. In those cases for which in addition to all of this automatic continuity is known to hold, such as when $M$ is c…

Surfaces and other Peano Continua with no Generic Chains
Gianluca Basso, Alessandro Codenotti, Andrea Vaccaro
https://arxiv.org/abs/2403.08667 https://

Surfaces and other Peano Continua with no Generic Chains
The space of chains on a compact connected space encodes all the different ways of continuously growing out of a point until exhausting the space. A chain is generic if its orbit under the action of the underlying homeomorphism group is comeager: a space X has a generic chain if there is essentially just one type of chain on X. Gutman, Tsankov and Zucker proved that compact manifolds of dimension at least 3 do not have a generic chain. We extend and generalize their result, covering a large cla…

This https://arxiv.org/abs/2404.02341 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Fully chaotic conservative models for some torus homeomorphisms
We study homotopic-to-the-identity torus homeomorphisms, whose rotation set has nonempty interior. We prove that any such map is monotonically semiconjugate to a homeomorphism that preserves the Lebesgue measure, and that has the same rotation set. Furthermore, the dynamics of the quotient map has several interesting chaotic traits: for instance, it is topologically mixing, has a dense set of periodic points and is continuum-wise expansive. In particular, this shows that a convex compact set of…

This https://arxiv.org/abs/2209.11257 has been replaced.
link: https://scholar.google.com/scholar?q=a

Linear actions of $\mathbb{Z}/p\times\mathbb{Z}/p$ on $S^{2n-1}\times S^{2n-1}$
For an odd prime $p$, we consider free actions of $(\mathbb{Z}/p)^2$ on $S^{2n-1}\times S^{2n-1}$ given by linear actions of $(\mathbb{Z}/p)^2$ on $\mathbb{R}^{4n}$. Simple examples include a lens space cross a lens space, but $k$-invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the $k$-invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demon…

Maximal Entropy Measures for Non-Accessible Topological Skew Products
Ali Tahzibi, Richard Cubas
https://arxiv.org/abs/2404.03169 https://

Maximal Entropy Measures for Non-Accessible Topological Skew Products
In this paper we establish a dichotomy for the ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with one-dimensional compact center leaves which are virtually skew products over (transitive) Anosov homeomorphism. We prove that if the whole manifold is the unique minimal invariant set saturated by unstable foliation, then either there exists a unique measure of maximal entropy which is non-hyperbolic or there are exactly two hyperbolic ergodic measures of maximal entr…

Density results for the modular group of infinite-type surfaces
Yassin Chandran, Tommaso Cremaschi
https://arxiv.org/abs/2403.20242 https://

Density results for the modular group of infinite-type surfaces
In this work we show two results about approximating, with respect to the compact-open topology, mapping classes on surfaces of infinite-type by quasi-conformal maps, in particular we are interested in density results. The first result is that given any infinite-type surface $S$ there exists a hyperbolic structure $X$ on $S$ such that $\text{PMCG}(S)\subseteq \overline{\text{Mod}(X)}$, for $\text{Mod}(X)$ the set of quasi-conformal homeomorphism on $X$. The second result is that given any surfa…

Fully chaotic conservative models for some torus homeomorphisms
Alejo Garc\'ia-Sassi, F\'abio Armando Tal
https://arxiv.org/abs/2404.02341 https://…

Fully chaotic conservative models for some torus homeomorphisms
We study homotopic-to-the-identity torus homeomorphisms, whose rotation set has nonempty interior. We prove that any such map is monotonically semiconjugate to a homeomorphism that preserves the Lebesgue measure, and that has the same rotation set. Furthermore, the dynamics of the quotient map has several interesting chaotic traits: for instance, it is topologically mixing, has a dense set of periodic points and is continuum-wise expansive. In particular, this shows that a convex compact set of…

This https://arxiv.org/abs/2312.16072 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Strong Nielsen equivalence on the punctured disc
Let $f$ be an orientation-preserving homeomorphism of the 2-disc $\mathbb{D}^2$ that fixes the boundary pointwise and leaves invariant a finite subset in the interior of $\mathbb{D}^2$. We study the strong Nielsen equivalence of periodic points of such homeomorphisms $f$ and we give a necessary and sufficient condition for two periodic points to be strong Nielsen equivalent in the context of braid theory. In addition, we present an application of our result to the trace formula given by Jiang--…