We recently found a kind of orbit equivalence relations, denoted by $E(G)$, which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^\omega /c(G)$, where $c(G)$ is the group of all convergent sequences in G. We list some main results.

Let G and H be two Polish groups. (1) G is a discrete countable group containing at least two elements iff $E(G)\sim _BE_0$; (2) G is non-archimedean iff $E(G)\le _B=^+$; (3) if H is CLI but G is not, then $E(G)\not \le _BE_H^Y$ for any Polish H-space Y; (4) if H is a non-archimedean Polish group but G is not, then $E(G)\not \le _BE_H^Y$ for any Polish H-space Y; (5) if H is TSI but G is not, then $E(G)\not \le _BE(H)$; (6) G is TSI non-archimedean iff $E(G)\leq _B E_0^\omega $ iff $E(G)\leq _B {\mathbb {R}}^\omega /c_0$; in particular, $E(G)\sim _B E_0^\omega $ iff G is TSI uncountable non-archimedean.

The notion of $\alpha $-l.m.-unbalanced Polish group for $\alpha <\omega _1$ is introduced. Let $G,H$ be Polish groups, $0<\alpha <\omega _1$. If G is $\alpha $-l.m.-unbalanced but H is not, then $E(G)\not \le _B E(H)$.

For TSI Polish groups, we proved a Pre-rigid Theorem: the existence of Borel reduction is transformed into the existence of a well-behaved continuous mapping between topological groups. As its applications, we proved several rigid theorems for locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively. In particular, we obtained a complete answer for abelian connected Lie groups: for $b_0,e_0,b_1,e_1\in {\mathbb {N}}$, $E({\mathbb {R}}^{b_0}\times {\mathbb {T}}^{e_0})\le _BE({\mathbb {R}}^{b_1}\times {\mathbb {T}}^{e_1})$ iff $e_0\le e_1$ and $b_0+e_0\le b_1+e_1$. In addition, we also proved that, the partially ordered set $P(\omega )/\text {Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb {R}}^n)$ and $E({\mathbb {T}}^n)$.

Friedman and Stanley [9] developed the notion of Borel reducibility and illustrated its use in comparing classification problems for some familiar classes of countable structures. For many embeddings, the fact that the embedding is 1–1 on isomorphism types is explained by the existence of simple formulas that, uniformly, interpret the input structure in the output structure. For the embeddings of graphs in trees, and in linear orderings, there is no uniform interpretation [16, 20]. We focus on a version of the Friedman–Stanley embedding from [16] that takes each structure $\mathcal {A}$ for the language of graphs to a labeled tree $T_{\mathcal {A}}$. Gonzalez and Rossegger [13] showed that this embedding preserves Scott complexity. We refine this result, showing that for an X-computable ordinal, if one of $\mathcal {A}$, $T_{\mathcal {A}}$ has a computable infinitary Scott sentence, then so does the other, and the complexities match. Let $\mathbb {T}$ be the class of labeled trees isomorphic to those in the range of the embedding, and let $\mathbb {T}^\alpha $ be the subclass consisting of structures of Scott rank at most $\alpha $. It follows from results of Gao [10] that $\mathbb {T}$ is not Borel. We show that for each $\alpha $, $\mathbb {T}^\alpha $ is Borel. In fact, if $\alpha $ is an X-computable ordinal, then $\mathbb {T}^\alpha $ is complete X-effective $\Pi _{2\alpha +2}$.

In this article, we deal with the classification complexity of continuous (Devaney) chaotic systems in dimensions $0,1,$ and $\infty $ using the framework of invariant descriptive set theory. We identify the complexity in dimensions $0$ and $\infty $, while in dimension $1$ we get some partial results.

More precisely, we prove the topological conjugacy relation of invertible chaotic systems on the Hilbert cube (resp. on all compact metric spaces) has the same complexity as (i.e., is Borel bireducible with) the universal orbit relation induced by a Polish group. As a consequence, this answers a recent question asked by L. Ding. We also prove that the topological conjugacy relation of invertible chaotic systems on the Cantor space has the same complexity as the universal relation induced by the group $S_\infty $. This answers a recent question by M. Foreman. Some non-trivial bounds on the classification complexity of chaotic systems on the interval and on the circle are also obtained. Namely, the lower bound is the Vitali equivalence relation, and the upper bound is the equality of countable sets of reals. This especially implies that the relation is Borel. However, the exact complexity remains unknown.

The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^\omega /c(G)$, where $c(G)$ is the group of all convergent sequences in G. Let G be a Polish group. (1) G is a discrete countable group containing at least two elements iff $E(G)\sim _BE_0$; (2) if G is TSI uncountable non-Archimedean, then $E(G)\sim _BE_0^\omega $; (3) G is non-Archimedean iff $E(G)\le _B=^+$; (4) if H is a CLI Polish group but G is not, then $E(G)\not \le _BE(H)$; and (5) if H is a non-Archimedean Polish group but G is not, then $E(G)\not \le _BE(H)$. The notion of $\alpha $-l.m.-unbalanced Polish group for $\alpha <\omega _1$ is introduced. Let G and H be Polish groups, $0<\alpha <\omega _1$. If G is $\alpha $-l.m.-unbalanced but H is not, then $E(G)\not \le _B E(H)$. For TSI Polish groups, the existence of Borel reduction is transformed into the existence of a well-behaved continuous mapping between topological groups. As its applications, for any Polish group G, let $G_0$ be the connected component of the identity element $1_G$. Let G and H be two separable TSI Lie groups. If $E(G)\le _BE(H)$, then there exists a continuous locally injective map $S:G_0\to H_0$. Moreover, if $G_0$ and $H_0$ are abelian, S is a group homomorphism. In particular, for $c_0,e_0,c_1,e_1\in {\mathbb {N}}$, $E({\mathbb {R}}^{c_0}\times {\mathbb {T}}^{e_0})\le _BE({\mathbb {R}}^{c_1}\times {\mathbb {T}}^{e_1})$ iff $e_0\le e_1$ and $c_0+e_0\le c_1+e_1$.

Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^\omega /c(G)$, where $c(G)$ is the group of all convergent sequences in G. We first established two results:

1. (1) Let $G,H$ be two Polish groups. If H is TSI but G is not, then $E(G)\not \le _BE(H)$.

2. (2) Let G be a Polish group. Then the following are equivalent: (a) G is TSI non-archimedean; (b)$E(G)\leq _B E_0^\omega $; and (c) $E(G)\leq _B {\mathbb {R}}^\omega /c_0$. In particular, $E(G)\sim _B E_0^\omega $ iff G is TSI uncountable non-archimedean.

A critical theorem presented in this article is as follows: Let G be a TSI Polish group, and let H be a closed subgroup of the product of a sequence of TSI strongly NSS Polish groups. If $E(G)\le _BE(H)$, then there exists a continuous homomorphism $S:G_0\rightarrow H$ such that $\ker (S)$ is non-archimedean, where $G_0$ is the connected component of the identity of G. The converse holds if G is connected, $S(G)$ is closed in H, and the interval $[0,1]$ can be embedded into H.

As its applications, we prove several Rigid theorems for TSI Lie groups, locally compact Polish groups, separable Banach spaces, and separable Fréchet spaces, respectively.

Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^{\omega }/c(G)$, where $c(G)$ is the group of all convergent sequences in G. The connected component of the identity of a Polish group G is denoted by $G_0$.

Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq _B E(H)$, then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker (S)$ is non-archimedean. The converse is also true when G is connected and compact.

For $n\in {\mathbb {N}}^+$, the partially ordered set $P(\omega )/\mbox {Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb {R}}^n)$ and $E({\mathbb {T}}^n)$.

We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of Hjorth’s conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

We prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts, which strengthens and simplifies recent results of Chang and Gao, and Cieśla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel bireducible with the universal orbit equivalence relation of the permutation group on a countable set. On the other hand we prove that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures.

In this article, a notion of Schauder equivalence relation ℝℕ/L is introduced, where L is a linear subspace of ℝℕ and the unit vectors form a Schauder basis of L. The main theorem is to show that the following conditions are equivalent:

(1) the unit vector basis is boundedly complete;

(2) L is a F σ in ℝℕ;

(3) ℝℕ/L is Borel reducible to ℓ ∞.

We show that any Schauder equivalence relation generalized by a basis of ℓ 2 is Borel bireducible to ℝℕ/ℓ 2 itself, but it is not true for bases of c 0 or ℓ 1. Furthermore, among all Schauder equivalence relations generated by sequences in c 0, we find the minimum and the maximum elements with respect to Borel reducibility.

We present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory.