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Su Gao

Publications and source records attributed to Su Gao.

At least 37 records · Page 2Linked to original sources
arXiv2024

An order analysis of hyperfinite Borel equivalence relations

In this paper we first consider hyperfinite Borel equivalence relations with a pair of Borel $\mathbb{Z}$-orderings. We define a notion of compatibility between such pairs, and prove a dichotomy theorem which characterizes exactly when a pair of Borel $\mathbb{Z}$-orderings are compatible with each other. We show that, if a pair of Borel $\mathbb{Z}$-orderings are incompatible, then a canonical incompatible pair of Borel $\mathbb{Z}$-orderings of $E_0$ can be Borel embedded into the given pair. We then consider hyperfinite-over-finite equivalence relations, which are countable Borel equivalence relations admitting Borel $\mathbb{Z}^2$-orderings. We show that if a hyperfinite-over-hyperfinite equivalence relation $E$ admits a Borel $\mathbb{Z}^2$-ordering which is self-compatible, then $E$ is hyperfinite.

math.LO↗
arXiv2024

Continuous Edge Chromatic Numbers of Abelian Group Actions

We prove that for any generating set $S$ of $\mathbb {Z}^n$, the continuous edge chromatic number of the Schreier graph of the Bernoulli shift action $G=F(S,2^{\mathbb{Z}^n})$ is $χ'_c(G)=χ'(G)+1$. In particular, for the standard generating set, the continuous edge chromatic number of $F(2^{\mathbb {Z}^n})$ is $2n+1$.

math.CO↗
arXiv2024

On finite spacer rank for words and subshifts

We define a notion of rank for words and subshifts that we call spacer rank, extending the notion of rank-one symbolic shifts of Gao and Hill. We construct infinite words of each finite spacer rank, of unbounded spacer rank, and show there exist words that do not have a spacer rank construction. We consider words that are fixed points of substitutions and give explicit conditions for the word to have an at most spacer rank two construction, and not to be rank one. We prove that finite spacer rank subshifts have topological entropy zero, and that there are zero entropy subshifts not defined by a word with a finite spacer rank construction. We also study shift systems associated with infinite words, including those associated to Sturmian sequences, which we show are spacer rank-two systems.

math.DS↗
arXiv2024

Subshifts of finite symbolic rank

The definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank-$1$ subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts we also show that the rank-1 subshifts are dense but not generic. Finally we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank $2$, and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.

math.DS↗
arXiv2024

Borel Combinatorics of Abelian Group Actions

We study the free part of the Bernoulli action of $\mathbb{Z}^n$ for $n\geq 2$ and the Borel combinatorics of the associated Schreier graphs. We construct orthogonal decompositions of the spaces into marker sets with various additional properties. In general, for Borel graphs $Γ$ admitting weakly orthogonal decompositions, we show that $χ_B(Γ)\leq 2χ(Γ)-1$ under some mild assumptions. As a consequence, we deduce that the Borel chromatic number for $F(2^{\mathbb{Z}^n})$ is $3$ for all $n\geq 2$. Weakly orthogonal decompositions also give rise to Borel unlayered toast structures. We also construct orthogonal decompositions of $F(2^{\mathbb{Z}^2})$ with strong topological regularity, in particular with all atoms homeomorphic to a disk. This allows us to show that there is a Borel perfect matching for $F(2^{\mathbb{Z}^n})$ for all $n\geq 2$ and that there is a Borel lining of $F(2^{\mathbb{Z}^2})$.

math.LO↗
arXiv2023

Continuous Combinatorics of Abelian Group Actions

This paper develops techniques which are used to answer a number of questions in the theory of equivalence relations generated by continuous actions of abelian groups. The methods center around the construction of certain specialized hyper-aperiodic elements, which produce compact subflows with useful properties. For example, we show that there is no continuous $3$-coloring of the Cayley graph on $F(2^{\mathbb{Z}^2})$, the free part of the shift action of $\mathbb{Z}^2$ on $2^{\mathbb{Z}^2}$. With earlier work of the authors this computes the continuous chromatic number of $F(2^{\mathbb{Z}^2})$ to be exactly $4$. Combined with marker arguments for the positive directions, our methods allow us to analyze continuous homomorphisms into graphs, and more generally equivariant maps into subshifts of finite type. We present a general construction of a finite set of "tiles" for $2^{\mathbb{Z}^n}$ (there are $12$ for $n=2$) such that questions about the existence of continuous homomorphisms into various structures reduce to finitary combinatorial questions about the tiles. This tile analysis is used to deduce a number of results about $F(2^{\mathbb{Z}^n})$.

math.LO↗
arXiv2023

The amalgamation property and Urysohn structures in continuous logic

In this paper we consider the classes of all continuous $\mathcal{L}$-(pre-)structures for a continuous first-order signature $\mathcal{L}$. We characterize the moduli of continuity for which the classes of finite, countable, or all continuous $\mathcal{L}$-(pre-)structures have the amalgamation property. We also characterize when Urysohn continuous $\mathcal{L}$-(pre)-structures exist, establish that certain classes of finite continuous $\mathcal{L}$-structures are countable Fraïssé classes, prove the coherent EPPA for these classes of finite continuous $\mathcal{L}$-structures, and show that actions by automorphisms on finite $\mathcal{L}$-structures also form a Fraïssé class. As consequences, we have that the automorphism group of the Urysohn continuous $\mathcal{L}$-structure is a universal Polish group and that Hall's universal locally finite group is contained in the automorphism group of the Urysohn continuous $\mathcal{L}$-structure as a dense subgroup.

math.LO↗
arXiv2021

Sarnak's conjecture for rank-one subshifts

Using techniques developed in \cite{KLR}, we verify Sarnak's conjecture for two classes of rank-one subshifts with unbounded cutting parameters. The first class of rank-one subshifts we consider are called {\em almost complete congruency classes} ({\em accc}), the definition of which is motivated by the main result of \cite{GS}, which implies that, when a rank-one subshift carries a unique non-atomic invariant probability measure, it is accc if it is measure-theoretically isomorphic to an odometer. The second class we consider consists of Katok's map and its generalizations.

math.DS↗
arXiv2021

Rank-One Transformations, Odometers, and Finite Factors

In this paper we give explicit characterizations, based on the cutting and spacer parameters, of (a) which rank-one transformations factor onto a given finite cyclic permutation, (b) which rank-one transformations factor onto a given odometer, and (c) which rank-one transformations are isomorphic to a given odometer. These naturally yield characterizations of (d) which rank-one transformations factor onto some (unspecified) finite cyclic permutation, (d') which rank-one transformations are totally ergodic, (e) which rank-one transformations factor onto some (unspecified) odometer, and (f) which rank-one transformations are isomorphic to some (unspecified) odometer.

math.DS↗
arXiv2021

On extensions of partial isometries

In this paper we define a notion of S-extension for a metric space and study minimality and coherence of S-extensions. We show that every S-extension can be identified with an algebraic object. We use this algebraic representation to give a complete characterization of all finite minimal S-extensions of a given finite metric space and a complete characterization of all minimal coherent S-extensions. We also define a notion of ultraextensive metric spaces and show that every countable metric space can be extended to a countable ultraextensive metric space. %As an application, we show that every countable subset of the Urysohn metric space can be extended to a countable dense ultraextensive subset of the Urysohn space. We also show that the isometry group of an infinite ultraextensive metric space has a dense locally finite subgroup, generalizing several previously known results. We also study compact ultrametric spaces and show that every compact ultrametric space can be extended to a compact ultraextensive ultrametric space.

math.LO↗
arXiv2021

A Marching Cube Algorithm Based on Edge Growth

Marching Cube algorithm is currently one of the most popular 3D reconstruction surface rendering algorithms. It forms cube voxels through the input image, and then uses 15 basic topological configurations to extract the iso-surfaces in the voxels. It processes each cube voxel in a traversal manner, but it does not consider the relationship between iso-surfaces in adjacent cubes. Due to ambiguity, the final reconstructed model may have holes. We propose a Marching Cube algorithm based on edge growth. The algorithm first extracts seed triangles, then grows the seed triangles and reconstructs the entire 3D model. According to the position of the growth edge, we propose 17 topological configurations with iso-surfaces. From the reconstruction results, the algorithm can reconstruct the 3D model well. When only the main contour of the 3D model needs to be organized, the algorithm performs well. In addition, when there are multiple scattered parts in the data, the algorithm can extract only the 3D contours of the parts connected to the seed by setting the region selected by the seed.

cs.GR↗
arXiv2020

Dense locally finite subgroups of Automorphism Groups of Ultraextensive Spaces

We verify a conjecture of Vershik by showing that Hall's universal countable locally finite group can be embedded as a dense subgroup in the isometry group of the Urysohn space and in the automorphism group of the random graph. In fact, we show the same for all automorphism groups of known infinite ultraextensive spaces. These include, in addition, the isometry group of the rational Urysohn space, the isometry group of the ultrametric Urysohn spaces, and the automorphism group of the universal $K_n$-free graph for all $n\geq 3$. Furthermore, we show that finite group actions on finite metric spaces or finite relational structures form a Fraïssé class, where Hall's group appears as the acting group of the Fraïssé limit. We also embed continuum many non-isomorphic countable universal locally finite groups into the isometry groups of various Urysohn spaces, and show that all dense countable subgroups of these groups are mixed identity free (MIF). Finally, we give a characterization of the isomorphism type of the isometry group of the Urysohn $Δ$-metric spaces in terms of the distance value set $Δ$.

math.LO↗
arXiv2020

On Extensions of Partial Isomorphisms

In this paper we study a notion of HL-extension (HL standing for Herwig--Lascar) for a structure in a finite relational language $\mathcal{L}$. We give a description of all finite minimal HL-extensions of a given finite $\mathcal{L}$-structure. In addition, we study a group-theoretic property considered by Herwig--Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal{L}$-structures and show that every countable $\mathcal{L}$-structure can be extended to a countable ultraextensive structure. Finally, it follows from our results that the automorphism group of any countable ultraextensive $\mathcal{L}$-structure has a dense locally finite subgroup.

math.LO↗
arXiv2019

Topological Factors of Rank-One Subshifts

We study topological factors of rank-one subshifts and prove that those factors that are themselves subshifts are either finite or isomorphic to the original rank-one subshifts. Thus, we completely characterize the subshift factors of rank-one subshifts.

math.DS↗
arXiv2019

The Complexity of the Classification Problems of Finite-Dimensional Continua

We consider the homeomorphic classification of finite-dimensional continua as well as several related equivalence relations. We show that, when $n \geq 2$, the classification problem of $n$-dimensional continua is strictly more complex than the isomorphism problem of countable graphs. We also obtain results that compare the relative complexity of various equivalence relations.

math.LO↗
arXiv2018

Topological Mixing Properties of Rank-One Subshifts

We study topological mixing properties and the maximal equicontinuous factor of rank-one subshifts as topological dynamical systems. We show that the maximal equicontinuous factor of a rank-one subshift is finite. We also determine all the finite factors of a rank-one shift with a condition involving the cutting and spacer parameters. For rank-one subshifts with bounded spacer parameter we completely characterize weak mixing and mixing. For rank-one subshifts with unbounded spacer parameter we prove some sufficient conditions for weak mixing and mixing. We also construct some examples showing that the characterizations for the bounded spacer parameter case do not generalize to the unbounded spacer parameter case.

math.DS↗