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In this paper we address the question: How many pairwise non-isomorphic extremely amenable groups are there which are seprable metrizable or even Polish? We show that there are continuum many such groups. In fact we construct continuum many pairwise non-isomorphic extremely amenable groups as automorphism groups of countable structures. We also consider this classicfication problem from the point of view of descriptive set theory by showing that the class of all extremely amenable closed subgroups of $S_\infty$ is Borel and by obtaining some lower bounds for their isomorphism relation in the Borel reducibility hierarchy.

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 $\chi'_c(G)=\chi'(G)+1$. In particular, for the standard generating set, the continuous edge chromatic number of $F(2^{\mathbb {Z}^n})$ is $2n+1$.

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.

We define some natural notions of strong and weak Borel Ramsey properties for countable Borel equivalence relations and show that they hold for a countable Borel equivalence relation if and only if the equivalence relation is smooth. We also consider some variation of the notion for hyperfinite non-smooth Borel equivalence relations.

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 $\Gamma$ admitting weakly orthogonal decompositions, we show that $\chi_B(\Gamma)\leq 2\chi(\Gamma)-1$ 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 $F(2^{\mathbb{Z}^2})$.

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.

This file is composed of questions that emerged or were of interest during the workshop "Interactions between Descriptive Set Theory and Smooth Dynamics" that took place in Banff, Canada on 2022.

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 Fra\"iss\'e 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 Fra\"iss\'e 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.

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., Comment: 12 pages

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.

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., Comment: Changed the title to now use "spacer rank". Updated the abstract. Added some references. Small corrections. Shortened parts of sections 3 and 4 (please refer to older version for omitted details)

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 Fra\"iss\'{e} class, where Hall's group appears as the acting group of the Fra\"iss\'{e} 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 $\Delta$-metric spaces in terms of the distance value set $\Delta$., Comment: 37 pages

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.

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.

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., Comment: 20 pages

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.

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.

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 $F(2^{\mathbb{Z}^n})$., Comment: 126 pages, 47 figures