2.1 Subshifts

Fix a finite set $\mathcal {A}$ of cardinality $|{\mathcal {A}}| \in \mathbb {N}_{>1}$ called the alphabet and let $\mathcal {A}^{\star }=\bigcup _{n=1}^{\infty } \mathcal {A}^n$ be the set of finite words over $\mathcal {A}$ . The set of bi-infinite sequences $\mathcal {A}^{\mathbb {Z}}$ with the product topology is a compact space. A basis of neighborhoods for the topology is given by the clopen sets $\mathrm {Cyl}_k(w)= \{x\in \mathcal {A}^{\mathbb {Z}} \colon x_{[k,k+n-1]}=w\}$ for $k \in \mathbb {Z}$ and $w\in \mathcal {A}^n$ . It is metrized by

$$ \begin{align*}d(x,x') = 2^{-\inf\{|k|\in \mathbb{N} \colon x_{k} \ne x^{\prime}_{k}\}}.\end{align*} $$

The shift $\operatorname {\mathrm {\sigma }} \colon \mathcal {A}^{\mathbb {Z}} \to \mathcal {A}^{\mathbb {Z}}$ defined by $\operatorname {\mathrm {\sigma }}(x)_{n} = x_{n+1}$ is a homeomorphism.

A subsystem $(X,\operatorname {\mathrm {\sigma }})$ of $(\mathcal {A}^{\mathbb {Z}}, \operatorname {\mathrm {\sigma }})$ is called a subshift on $|{\mathcal {A}}|$ symbols. A word $w\in \mathcal {A}^{\star }$ is a subword of $x\in X$ when there exists $k\in \mathbb {Z}$ such that $x\in \mathrm {Cyl}_k(w)$ . The language of $(X,\operatorname {\mathrm {\sigma }})$ is the set $\mathcal {A}^{\star }_X \subset \mathcal {A}^{\star }$ of all subwords of all elements. There is an equivalence between subshifts of $(\mathcal {A}, \operatorname {\mathrm {\sigma }})$ and languages in $\mathcal {A}^\star $ that are factorial (closed by taking subwords) and extendable (every word is a proper subword of some other word). The factor complexity of $(X,\operatorname {\mathrm {\sigma }})$ is the function $C_X\colon \mathbb {N}\to \mathbb {N}$ defined by $C_X(n)=|{\mathcal {A}^{\star }_X \cap \mathcal {A}^n}|$ .

The subshift $(X,\operatorname {\mathrm {\sigma }})$ is called uniformly recurrent if for every $x\in X$ , every word in $\mathcal {A}^\star _X$ occurs in the sequence $\operatorname {\mathrm {\sigma }}^k(x)$ infinitely often and with bounded gaps. This is equivalent to the minimality of $(X,\operatorname {\mathrm {\sigma }})$ .

The subshift $(X,S)$ has a recurrence function $R_X\colon \mathbb {N}\to \mathbb {N}$ that to $n\in \mathbb {N}$ , associates the minimum $R(n)$ of the integers $r\in \mathbb {N}$ such that every word in $\mathcal {A}^r_X$ contains every word in $\mathcal {A}^n_X$ . One may show (see [Reference Berthé and DelecroixBD14, Theorem 2.3]) that $C_X\le R_X$ .

A subshift $(X,S)$ is called linearly recurrent when there exists $C\in \mathbb {R}_{>0}$ such that for all $n\in \mathbb {N}$ , we have $R_X(n)\le Cn$ . Linear recurrence implies uniform recurrence (equivalently minimality), sublinear factor complexity (as $C_X\le R_X$ ), and unique ergodicity (this is [Reference Berthé and DelecroixBD14, Theorem 2.5], which refers to [Reference Durand, Host and SkauDHS99]).

Let us recall a criterion due to Boshernitzan for linear recurrence of minimal subshifts (see [Reference DelecroixDel15, Theorem 21] for a proof). For a subshift $(X, \operatorname {\mathrm {\sigma }})$ with an invariant probability measure $\mu \in \mathcal {M}_{p}(X,\operatorname {\mathrm {\sigma }})$ , define the function $\epsilon _\mu \colon \mathbb {N}\to \mathbb {R}$ by $\epsilon _\mu (n)=\min \{\mu (\mathrm {Cyl}_0(w)\cap X) \colon w \in \mathcal {A}^n_X \}$ (the minimal measure of cylinders of width n). The minimal subshift $(X,\operatorname {\mathrm {\sigma }})$ is linearly recurrent if and only if any $\mu \in \mathcal {M}_{p}(X,\operatorname {\mathrm {\sigma }})$ satisfies $\inf n\epsilon _\mu (n)>0$ . It follows that if a subshift is totally minimal and linearly recurrent, then it is totally linearly recurrent; hence, totally uniquely ergodic.

Let us finally mention that the notions of uniform recurrence and linear recurrence can be characterized in terms of S-adic representations (see [Reference Berthé and DelecroixBD14, §5]). In particular, it is shown in [Reference DurandDur98] that a subshift is linearly recurrent if and only if it is a strongly primitive and proper S-adic subshift.

2.2 Rotations and Sturmian systems

On the circle $\mathbb {T}=\mathbb {R}\bmod {\mathbb {Z}}$ , let $R_\alpha \colon \mathbb {T} \to \mathbb {T}$ denote the rotation by $\alpha \in \mathbb {T}$ . If $\alpha $ is rational, then $R_\alpha $ is periodic. If $\alpha $ is irrational, then $(\mathbb {T}, R_\alpha )$ is minimal and uniquely ergodic, with Lebesgue measure on $\mathbb {T}$ the unique invariant probability measure.

For an irrational $\alpha \in \mathbb {T}$ , we define the Sturmian system $(X_\alpha , \operatorname {\mathrm {\sigma }}_\alpha )$ as the simplest symbolic encoding of $R_\alpha $ , that is, a subshift on two symbols that factors onto $(\mathbb {T}, R_\alpha )$ . For this, we construct a point $x \in \{0,1\}^{\mathbb {Z}}$ by

$$ \begin{align*} x_{n} = \begin{cases} 0 & \text{if } n \alpha \in [0,1-\alpha),\\ 1 & \text{if } n \alpha \in [1 - \alpha,0), \end{cases} \end{align*} $$

and let $X_{\alpha }$ be the closure of the $\operatorname {\mathrm {\sigma }}$ -orbit of x in $\{0,1\}^{\mathbb {Z}}$ .

The language of $X_{\alpha }$ has complexity $C_{X_\alpha }(n) = n+1$ (which is the smallest possible among aperiodic sequences, by Morse and Hedlund [Reference Morse and HedlundMH40]). Conversely, every recurrent subshift of complexity $C_{X}(n) = n+1$ is obtained by the previous construction for a unique $\alpha \in \mathbb {T}$ .

It is well known [Reference Pytheas Fogg, Berthé, Ferenczi, Mauduit and SiegelFog02] that for irrational $\alpha \in \mathbb {T}$ , the system $(X_{\alpha },\operatorname {\mathrm {\sigma }}_{\alpha })$ is minimal and uniquely ergodic.

The irrational $\alpha \in \mathbb {R}\setminus \mathbb {Q}$ has a unique Euclidean continued fraction expansion:

$$ \begin{align*} \lfloor\,{a_0,a_1,\ldots}\,\rfloor=a_0+\frac{1}{a_1+\cdots} \quad \text{with } a_j \in \mathbb{N} \text{ and }\text{for all } j>0,\; a_j>0. \end{align*} $$

Thus, it is the fixed point of the infinite product of matrices $R^{a_0}L^{a_1}\cdots $ where

$$ \begin{align*} L = \bigg(\begin{matrix} 1 & 0\\ 1 & 1 \end{matrix}\bigg), \quad R = \bigg(\begin{matrix} 1 & 1\\ 0 & 1 \end{matrix}\bigg) \end{align*} $$

act by linear fractional transformations on $\mathbb {R}\mathbf {P}^1$ .

Such a sequence can be generated from any seed by applying its S-adic representation, which is determined by the continued fraction expansion of $\alpha $ (see [Reference Schmieding and SimonSS24]).

2.3 Homeomorphisms of the circle and Denjoy systems

Let us briefly recall, following [Reference Putnam, Schmidt and SkauPSS86], the main results of Poincaré [Reference PoincaréPoi85] concerning homeomorphisms of the circle and their rotation numbers, and of Markley [Reference MarkleyMar70] describing the invariants for conjugacy classes in $\operatorname {\mathrm {Homeo}}^+(\mathbb {T})$ of Denjoy homeomorphisms.

Let $F\in \operatorname {\mathrm {Homeo}}^+(\mathbb {T})$ be an orientation preserving homeomorphism of the circle. Choose a lift to the universal cover $\tilde {F}\colon \mathbb {R}\to \mathbb {R}$ : it is determined by $\lfloor \tilde {F}(0)\rfloor \in \mathbb {Z}$ as any two lifts differ by an integer. There exists $\tau (\tilde {F}) \in \mathbb {R}$ such that for all $x\in \mathbb {R}$ , we have $({1}/{n})({F}^n(x)-x)\to \tau (\tilde {F})$ , and the value $\tau (\tilde {F}) \in \mathbb {R}\bmod {\mathbb {Z}}$ does not depend on the lift: it is called the rotation number of F and denoted $\rho (F)\in \mathbb {T}$ .

The rotation number $\rho \colon \operatorname {\mathrm {Homeo}}^+(\mathbb {T})\to \mathbb {T}$ is invariant by conjugacy in $\operatorname {\mathrm {Homeo}}^+(\mathbb {T})$ . More generally, if $F_1, F_2\in \operatorname {\mathrm {Homeo}}^+(\mathbb {T})$ and there exists a continuous surjective map $H\colon \mathbb {T}\to \mathbb {T}$ satisfying $HF_{1} = F_{2}H$ , then $\rho (F_1)=\pm \rho (F_2)$ , where the sign depends on the orientation preserving or reversing behavior of H. Moreover, for all $n\in \mathbb {Z}$ , we have $\rho (F^n)=n\rho (F)$ .

If $\rho (F)$ is rational, then F has a periodic point and F is conjugate to $R_{\rho (F)}$ . If $\rho (F)$ is irrational, then every point $x\in \mathbb {T}$ has an orbit $\{F^n(x)\colon n\in \mathbb {Z}\}$ with the same cyclic order structure as $\{n \cdot \rho (F)\colon n\in \mathbb {Z}\}$ ; hence, F is semi-conjugate to $R_{\rho (F)}$ by a map $H\colon \mathbb {T} \to \mathbb {T}$ that is unique up to post-composing by a rotation.

We say that F is a Denjoy homeomorphism when it is semi-conjugate but not conjugate to $R_{\rho (F)}$ , that is, when $\rho (F)$ is irrational and the semi-conjugacy $H\colon \mathbb {T} \to \mathbb {T}$ is not a homeomorphism (in which case, it is necessarily orientation preserving).

Fix a Denjoy homeomorphism $F\in \operatorname {\mathrm {Homeo}}^+(\mathbb {T})$ and let $H\colon \mathbb {T} \to \mathbb {T}$ be a semi-conjugacy from F to $R_\alpha $ . The homeomorphism F is uniquely ergodic with invariant probability measure $dH$ . The support of $dH$ , denoted ${\mathcal {D}}_F\subset \mathbb {T}$ , is homeomorphic to a Cantor space, and the Denjoy system $({\mathcal {D}}_F, F)$ is the unique minimal F-invariant subsystem of $(\mathbb {T}, F)$ . The restriction of H to $ {\mathcal {D}}_{F}$ yields a factor $({\mathcal {D}}_{F}, F) \to (\mathbb {T}, R_{\rho (F)})$ .

The open set $\mathbb {T}\setminus {\mathcal {D}}_F$ is a disjoint union of open intervals $I_n\subset \mathbb {T}$ indexed by $\mathbb {N}$ , and H collapses the closure of each interval $I_n$ to a point. The map F uniquely determines that set of points $\{H(I_n)\colon n\in \mathbb {N}\}$ up to post-composition by a rotation: denote by $Q(F)$ its rotation class. The set $\{I_n\colon n\in \mathbb {N}\}$ with its cyclic order and F-orbit structures is isomorphic to the set $Q(F)$ with its cyclic order and $R_{\rho (F)}$ -orbit structures.

The Denjoy homeomorphism F is characterized up to topological conjugacy in $\operatorname {\mathrm {Homeo}}^+(\mathbb {T})$ by $(\rho (F), Q(F))$ and in $\operatorname {\mathrm {Homeo}}(\mathbb {T})$ by $\pm (\rho (F), Q(F))$ [Reference MarkleyMar70]. Similarly, the Denjoy system $({\mathcal {D}}_F, F)$ is characterized up to cyclic-order preserving topological conjugacy by $( \rho (F), Q(F) )$ and up to topological conjugacy by $\pm (\rho (F), Q(F))$ (see [Reference Putnam, Schmidt and SkauPSS86, Remark 3]). From now on, for a pair consisting of an irrational $\alpha \in \mathbb {T}$ and a countable $R_\alpha $ -invariant subset $A \subset \mathbb {T}$ containing the origin, we denote such a Denjoy homeomorphism by $(\mathbb {T}, F_{\alpha , A})$ and its minimal subsystem by $({\mathcal {D}}_{\alpha , A}, F_{\alpha , A})$ .

The following is a key fact for us. The first part is folklore (for a proof, see for instance [Reference MasuiMas09]).

Note that a Denjoy system factors onto a unique Sturmian system, given by its rotation number $\alpha \bmod {\mathbb {Z}}$ .

2.4 Interval exchange transformations and their subshifts

Fix our alphabet $\mathcal {A}=\{1,\ldots , d\}$ for $d\in \mathbb {N}_{\ge 1}$ . Consider a permutation $\sigma \colon \mathcal {A} \to \mathcal {A}$ and length parameters $\unicode{x3bb} \in (\mathbb {R}_{>0})^{\mathcal {A}}$ . We fix $\alpha _0=0$ , define the discontinuities $\alpha _k=\sum _{1}^k \unicode{x3bb} _j$ for $k\in \mathcal {A}$ , and consider the intervals $I_k=(\alpha _{k-1},\alpha _{k}]$ of lengths $\unicode{x3bb} _k=\alpha _k-\alpha _{k-1}$ .

The interval exchange transformation (IET) with permutation $\sigma $ and lengths $\unicode{x3bb} $ is the function $T\colon (\alpha _0,\alpha _d] \to (\alpha _0,\alpha _d]$ permuting the intervals $I_k= (\alpha _{k-1}, \alpha _{k}]$ according to $\sigma $ . It is given by the formula

$$ \begin{align*} T(x\in I_k) = (x-\alpha_{k-1}) + \sum_{l>k, \sigma(l)<\sigma(k)} \unicode{x3bb}_l \: - \sum_{l<k, \sigma(l)>\sigma(k)} \unicode{x3bb}_l. \end{align*} $$

Two IETs are called equivalent when they have the same permutation and their length parameters yield proportional vectors in $\mathbb {R}_{>0}^{\mathcal {A}}$ . Every equivalence class contains a unique representative with $\sum \unicode{x3bb} _k=1$ , defined on the interval $(0,1]$ .

The symbolic encoding of T is the subshift $(X,\operatorname {\mathrm {\sigma }}) \subset (\mathcal {A}^{\mathbb {Z}}, \operatorname {\mathrm {\sigma }})$ defined as follows. We define $X_o \subset (\alpha _0,\alpha _d]$ to be the complement of the orbits of all discontinuities: a point $x\in X_o$ yields a sequence of labels $l_j \in \mathcal {A}$ recording the intervals $I_{l_j}$ containing the $T^j(x)$ . This defines a map $X_o \to \mathcal {A}^{\mathbb {Z}}$ and the closure of its image is X, a subshift. Such a subshift $(X,\operatorname {\mathrm {\sigma }})$ will be called an interval exchange system (IES).

For $m\in \mathbb {Z}$ , the mth power $T^m$ is an interval exchange with discontinuities given by $\alpha ^{(m)}_k\in \bigcup _{n=1}^{m} T^{1-n}(\{\alpha _0,\ldots ,\alpha _d\})$ , of which there are at most $(d-1)\lvert m\rvert +1$ , with equality unless some discontinuities belong to the same orbit. The intervals $(\alpha ^{(m)}_{k}, \alpha ^{(m)}_{k+1}] \subset (\alpha _0,\alpha _d]$ whose endpoints are consecutive discontinuities for $T^m$ correspond to cylinders $\mathrm {Cyl}_0(w)\subset X$ of words $w\in \mathcal {A}_X^m$ .

We say that T satisfies Keane’s infinite distinct orbit condition (or has ido) when the past-orbits of the discontinuities $\{T^{-n}(\alpha _k)\colon n\in \mathbb {N}\}$ are all infinite and disjoint. This implies that the map $X_o \to X$ is injective and its inverse, which is partially defined on its image, extends to a unique map $X\to [\alpha _0,\alpha _d]$ .

If T has ido, then $(X,\operatorname {\mathrm {\sigma }})$ has complexity function $C_X(n)=(d-1)n+1$ and it is minimal (Keane [Reference KeaneKea75] showed that the ido condition implies that every T-orbit is dense). Conversely, if a minimal IES $(X,\operatorname {\mathrm {\sigma }})$ has $C_X(n)=(d-1)n+1$ , then every IET with that encoding has ido (so it makes sense to say whether a minimal IES has ido).

If a minimal subshift on d unordered letters of complexity $n\mapsto (d-1)n+1$ comes from an IES, then the order on its alphabet $\mathcal {A}$ and the permutation $\pi \colon \mathcal {A} \to \mathcal {A}$ are uniquely determined up to the action of either the reflection group $\mathbb {Z}/2$ generated by the order reversing involution $k\mapsto d+1-k$ or else the dihedral group $\mathbb {Z}/d\rtimes \mathbb {Z}/2$ obtained as its semi-direct product by the cyclic group $\mathbb {Z}/d$ generated by the cyclic order preserving permutation $k\mapsto k+1\bmod {d}$ .

Given a minimal subshift $(X,\operatorname {\mathrm {\sigma }})\subset (\mathcal {A},\operatorname {\mathrm {\sigma }})$ of complexity $C_X(n)=(d-1)n+1$ on the ordered alphabet $\mathcal {A}$ , one may decide from its language whether there exist a permutation $\pi \colon \mathcal {A}\to \mathcal {A}$ and lengths $\unicode{x3bb} \colon \mathcal {A} \to \mathbb {R}_{>0}$ yielding an IET with ido having that encoding [Reference Ferenczi and ZamboniFZ08]. When so, the permutation $\pi \colon \mathcal {A} \to \mathcal {A}$ is uniquely determined and it follows from [Reference YoccozYoc10, §8.1] that the possible length parameters $\unicode{x3bb} $ are in bijective correspondence with the invariant measures of $(X, \operatorname {\mathrm {\sigma }})$ .

The permutation $\pi $ is called irreducible when $\pi ^{-1}(\{1,\ldots , k\})=\{1,\ldots , k\} \iff k\in \{1,d\}$ ; this means that T does not restrict to an IET on a smaller interval. If T has ido, then $\pi $ is irreducible. If $\pi $ is irreducible and the rational relations between the lengths $\unicode{x3bb} _k$ are generated by $\sum \unicode{x3bb} _k=1$ , then T has ido. Moreover, Masur [Reference MasuiMas82] and Veech [Reference VeechVee82] showed that for every irreducible $\pi $ , almost all probability vectors $\unicode{x3bb} $ yield an IES that is uniquely ergodic.

Note that a Denjoy system $({\mathcal {D}}_{\alpha , A}, F_{\alpha , A})$ with $A\bmod {R_\alpha }$ of cardinal $d\in \mathbb {N}_{\ge 2}$ is topologically conjugate to an IES on d intervals and its permutation $\pi $ is a d-cycle, which preserves the cyclic order on $\mathcal {A}\bmod {d}$ in the sense that if $(i,j,k)$ are cyclically ordered, then $(\pi (i),\pi (j),\pi (k))$ is also cyclically ordered. Conversely, all such cyclic-order-preserving d-cycles arise from Denjoy systems. For instance, the Sturmian system $(X_\alpha , \operatorname {\mathrm {\sigma }}_\alpha )$ is the IES associated to the transposition $\tau $ of $d=2$ intervals with lengths $\unicode{x3bb} _1 = \alpha $ and $\unicode{x3bb} _2 = 1-\alpha $ .

3.1 Coinvariants, states, and cohomology of the suspension

Let $(X,S)$ be a system and denote by $C(X,\mathbb {Z})$ the abelian group of continuous integer valued functions on X. It is endowed with the coboundary homomorphism $f\mapsto f - f\circ S$ , whose image defines the subgroup of coboundaries $\textrm {cbd}_{S}(X,\mathbb {Z})$ and whose cokernel defines the group of coinvariants

$$ \begin{align*}{\mathcal{G}}_{S} = C(X,\mathbb{Z}) / \textrm{cbd}_{S}(X,\mathbb{Z}).\end{align*} $$

A homomorphism $\tau \colon {\mathcal {G}}_{S} \to \mathbb {R}$ is called positive if $\tau ([f])$ is non-negative whenever $[f]=[g]$ for some non-negative $g \in C(X,\mathbb {Z}_{\ge 0})$ , and a positive homomorphism $\tau $ is called a state when $\tau ([1]) = 1$ . We let $\mathcal {P}(X,S)$ denote the space of states for a system $(X,S)$ .

The following is very well known; a proof can be found in [Reference Boyle and HandelmanBH96, §1.6].

Proposition 3.1. (States as measures)

Let $(X,S)$ be a zero-dimensional system. The space of states is in bijection with the space $\mathcal {M}_{p}(X,S)$ of S-invariant Borel probability measures on X by the map sending ${\mu \in \mathcal {M}_{p}(X,S)}$ to the state $\tau _\mu \in \mathcal {P}(X,S)$ defined by

$$ \begin{align*}\tau_{\mu}(f) = \int_{X}f \, d\mu.\end{align*} $$

More precisely, the space of states and the space of S-invariant Borel probability measures on X both have the structure of a non-empty metrizable Choquet simplex. The map $\mu \mapsto \tau _\mu $ is an isomorphism, in particular, it preserves the convex structure.

A morphism of systems $\pi \colon (X,S)\to (Y,T)$ yields an induced homomorphism of groups $\pi ^*\colon {\mathcal {G}}_{T} \to {\mathcal {G}}_{S}$ given by $\pi ^{*}([f]) = f \circ \pi $ and an associated affine morphism of simplices $\pi _*\colon {\mathcal {P}}(X,S) \to {\mathcal {P}}(Y,T)$ given by $\pi _{*}(\tau )([f]) = \tau (\pi ^{*}([f]))$ . For minimal Cantor systems, the induced map $\pi ^*$ on coinvariants is injective and, hence, the map $\pi _*$ on states is surjective (see [Reference Glasner and WeissGW95, Proposition 3.1]).

We define the subgroup of infinitesimals $\operatorname {\mathrm {Inf}}_{S}\subset {\mathcal {G}}_{S}$ as the set of classes $[f] \in {\mathcal {G}}_{S}$ such that $\tau ([f]) = 0$ for every state $\tau $ on ${\mathcal {G}}_{S}$ , or equivalently, such that $\int _{X} f\, d\mu = 0$ for every S-invariant Borel probability measure $\mu $ on X; in formula,

$$ \begin{align*}\operatorname{\mathrm{Inf}}_{S} = \bigcap_{\tau \in \mathcal{P}(X,S)} \ker(\tau \colon {\mathcal{G}}_{S} \to \mathbb{R}).\end{align*} $$

Let $(X,S)$ be a system. The product space $X \times \mathbb {R}$ is endowed with the diagonal action of the lattice $\mathbb {Z}\subset \mathbb {R}$ by $n\cdot (x,s)=(S^{n}(x),s-n)$ . The quotient $\Sigma _S X := X\times \mathbb {R} \bmod {\mathbb {Z}}$ is endowed with the $\mathbb {R}$ -action by left translation of the right component, namely the flow $\Phi _{s^{\prime }}(x,s)=(x,s+s^{\prime })$ . We call $\Sigma _S X$ the suspension of the system $(X,S)$ .

Note that if $(M,S)$ is a minimal subsystem of $(X,S)$ , then $\Sigma _S M$ is a minimal subsystem of $\Sigma _S X$ . Conversely, if N is a minimal subsystem of $\Sigma _S X$ , then the set $M=\{x\in X\colon (x,0)\in N\}$ yields a minimal subsystem $(M,S)$ of $(X,S)$ . Moreover, the projection $\Sigma _S X \to X$ yields an isomorphism $\mathcal {P}(\Sigma _S X,\Phi ) \to \mathcal {P}(X,S)$ between the respective spaces of invariant Borel probability measures (see [Reference Itzá-OrtizIO15, Lemma 3.8]).

For a topological space Y, let $\pi ^{1}(Y)$ denote the group of homotopy classes of maps from Y to the unit circle $\mathbf {S}^{1}$ , for which the group operation is pointwise multiplication in the target $\mathbf {S}^1$ . If Y is compact Hausdorff, then $\pi ^{1}(Y)$ is isomorphic to the first integral Čech cohomology group $\check {H}^{1}(Y,\mathbb {Z})$ .

Now, suppose that $(X,S)$ is a zero-dimensional system. Given a map ${\varphi \colon \Sigma _{S}X \to \mathbf {S}^{1}}$ , there exists for every $z \in \Sigma _{S}X$ a continuous function $\phi _{z} \colon \mathbb {R} \to \mathbb {R}$ satisfying $\phi _z(0)=0$ and $\varphi (z+s) = \varphi (z)e^{2 \pi i \phi _{z}(s)}$ for all $s \in \mathbb {R}$ . Denote by $C_{+}(\Sigma _{S}X,\mathbf {S}^{1})$ the set of maps $\varphi \colon \Sigma _{S}X \to \mathbf {S}^{1}$ such that we may choose $\phi _{z}$ to be non-decreasing for all $z \in \Sigma _{S}X$ . The abelian group $\pi ^{1}(\Sigma _{S}X)$ is endowed with the pre-order defined by the positive cone

$$ \begin{align*} \pi^{1}_{+}(\Sigma_{S}X) = \{ [\varphi] \colon \varphi \in C_{+}(\Sigma_{S}X,\mathbf{S}^{1})\} \subset \pi^{1}(\Sigma_{S}X). \end{align*} $$

For $f \in C(X,\mathbb {Z})$ , we define $\varphi _{f} \in C(\Sigma _{S}X,\mathbf {S}^{1})$ by

$$ \begin{align*} \varphi_{f}(x,s) = e^{2 \pi i s f(x)}, \quad 0 \le s < 1. \end{align*} $$

The following result is well known and a proof may be found in [Reference Boyle and HandelmanBH96, Proposition 4.5].

Proposition 3.2. (From coinvariants to homotopy classes)

Let $(X,S)$ be a zero-dimensional system. The exponential map yields an isomorphism of groups:

(1) $$ \begin{align} \begin{gathered} {\mathcal{G}}_{S} \to \pi^{1}(\Sigma_{S}X)\\ [f] \longmapsto [\varphi_{f}]. \end{gathered} \end{align} $$

Moreover, if $(X,S)$ is minimal, then the isomorphism respects the order structures and, hence, gives an isomorphism of ordered abelian groups

$$ \begin{align*} ({\mathcal{G}}_{S},{\mathcal{G}}_{S}^{+}) \cong (\pi^{1}(\Sigma_{S}X),\pi^{1}_{+}(\Sigma_{S}X)). \end{align*} $$

3.2 Coinvariants of Denjoy and interval exchange systems

Recall from §2.3 the definition of Denjoy homeomorphisms as well as the invariants classifying them up to (orientation preserving) conjugacy in $\operatorname {\mathrm {Homeo}}(\mathbb {T})$ .

Fix an irrational $\alpha \in \mathbb {T}$ and an $R_\alpha $ -invariant countable subset $A\subset \mathbb {T}$ containing the origin. Let $N\in \mathbb {N}\sqcup \{\infty \}$ denote the number of $R_{\alpha }$ -orbits in A and choose a point $a_i\in A$ in each of these orbits in such a way that $a_1=\alpha $ . Such pairs $(\alpha , A)$ parametrize Denjoy homeomorphisms $(\mathbb {T}, F_{\alpha , A})$ and Denjoy systems $({\mathcal {D}}_{\alpha , A}, F_{\alpha , A})$ up to cyclic order preserving conjugacy.

The surjective map $H\colon \mathbb {T} \to \mathbb {T}$ yielding the factor $(\mathbb {T}, F_{\alpha , A}) \to (\mathbb {T}, R_{\alpha })$ restricts to a factor $\pi _{\alpha , A} \colon ({\mathcal {D}}_{\alpha , A}, F_{\alpha , A}) \to (\mathbb {T}, R_{\alpha })$ . Both systems $(\mathbb {T}, F_{\alpha , A})$ and $({\mathcal {D}}_{\alpha , A}, F_{\alpha , A})$ are uniquely ergodic, with invariant probability measures given by $dH$ on $\mathbb {T}$ and its restriction to its support ${\mathcal {D}}_{\alpha , A}$ .

The following is [Reference Putnam, Schmidt and SkauPSS86, Theorem 5.3].

Let us briefly outline some of the ideas in the proof, as it will be useful later.

We now turn to the symbolic codings of IETs satisfying Keane’s ido condition. Their coinvariants and trace follow from [Reference PutnamPut89] or [Reference PutnamPut92].

4.1 Flow equivalence and states

A flow equivalence between two systems $(X,S)$ and $(Y,T)$ is a homeomorphism between their suspensions $\Sigma _{S}X \to \Sigma _{T}Y$ that maps flow orbits bijectively to flow orbits and preserves the orientation on each flow orbit. Two systems are called flow equivalent when there exists a flow equivalence between their suspensions.

Note that a topological conjugacy $\psi \colon (X,S) \to (Y,T)$ induces a flow equivalence ${\Psi \colon \Sigma _{S}X \to \Sigma _{T}Y}$ defined by $\Psi \colon (x,t) \mapsto (\psi (x),t)$ .

Now, consider zero-dimensional systems $(X,S)$ and $(Y,T)$ with a flow equivalence ${\Psi \colon \Sigma _{S}X \to \Sigma _{T}Y}$ . The map $\Psi $ induces an isomorphism of groups

$$ \begin{align*} \begin{gathered} \Psi^{*} \colon \pi^{1}(\Sigma_{T}Y) \to \pi^{1}(\Sigma_{S}X)\\ \Psi^{*}([\varphi]) = [\varphi \circ \Psi], \end{gathered} \end{align*} $$

which preserves the positive cones, since $\Psi $ preserves the orientation induced by the flow. Hence, it follows from Proposition 3.2 that the flow equivalence $\Psi \colon \Sigma _{S}X \to \Sigma _{T}Y$ also induces an isomorphism $\Psi ^{*} \colon {\mathcal {G}}_{T} \to {\mathcal {G}}_{S}$ . When $(X,S)$ is minimal, the standard isomorphism given in Proposition 3.2 respects the order (see [Reference Boyle and HandelmanBH96, Proposition 4.5]); thus, $\Psi ^{*} \colon {\mathcal {G}}_{T} \to {\mathcal {G}}_{S}$ is an isomorphism of ordered groups. An explicit method to compute $\Psi ^{*}$ without appealing to Proposition 3.2 can be found in [Reference Schmieding and YangSY21, §4]. Note that, in general, $\Psi ^{*}([1])$ need not be equal to $[1]$ .

When $(X,S)$ and $(Y,T)$ are minimal, the induced isomorphism $\Psi ^{*} \colon {\mathcal {G}}_{T} \to {\mathcal {G}}_{S}$ that is order preserving thus induces a linear isomorphism between the spaces of positive homomorphisms

(2) $$ \begin{align} \begin{gathered} \Psi_{*} \colon \mathcal{P}({\mathcal{G}}_{S}) \to \mathcal{P}({\mathcal{G}}_{T})\\ \Psi_{*}(\tau)(g) = \tau(\Psi^{*}(g)). \end{gathered} \end{align} $$

We summarize this in the following proposition.

4.2 Flow equivalence and discrete cross sections

Let $(X,S)$ be a zero-dimensional system. A discrete cross section of $(X,S)$ is a closed subset $C \subset X$ such that ${X = \{S^{i}(x) \colon x \in C, i \in \mathbb {N}_{>0}\}}$ and the function $r_{C} \colon C \to \mathbb {N}_{>0}$ defined by $r_{C}(x) = \min \{i \in \mathbb {N}_{>0} \colon S^{i}(x) \in C\}$ is continuous. Note that the continuity of $r_C$ implies that C will be clopen. Associated to a discrete cross section $C \subset X$ is a return system $(C,S_{C})$ defined by

$$ \begin{align*} \begin{gathered} S_{C} \colon C \to C\\ S_{C} \colon x \mapsto S^{r_{C}(x)}x. \end{gathered} \end{align*} $$

The systems $(X,S)$ and $(C,S_{C})$ are always flow equivalent and there is a natural flow equivalence $\Psi \colon \Sigma _{S}X \to \Sigma _{S_C} C$ defined as follows. For each $x \in C$ , $\Psi $ maps the segment in $\Sigma _{S}X$ with endpoints $(x,0)$ and $(x,r_C(x)) = (S_{C}(x),0)$ linearly via ${t \mapsto ({1}/{r_{C}(x)})t}$ onto the segment in $\Sigma _{S_{C}}C$ with endpoints $(x,0)$ and $(x,1) = (S_{C}(x),0) \in \Sigma _{S_{C}}X$ . That $\Psi $ is well defined and is a flow equivalence then follows from the fact that C is a discrete cross section of $(X,S)$ .

Note that if $(X,S)$ is minimal, then any clopen subset is a discrete cross section and any return system $(C,S_{C})$ is also minimal.

It follows from the above discussion that if $(C,S_{C})$ and $(D,T_{D})$ are return systems for $(X,S)$ and $(Y,T)$ , then any topological conjugacy $\psi \colon (C,S_{C}) \to (D,T_{D})$ induces a flow equivalence between their suspensions, and hence, a flow equivalence between the suspensions of the initial systems

$$ \begin{align*} \begin{gathered} \Psi \colon \Sigma_{S}X \to \Sigma_{T}Y \\ \Psi \colon (x,t) \mapsto (\psi(x),t). \end{gathered} \end{align*} $$

A return equivalence between systems $(X,S)$ and $(Y,T)$ is a triple $(\psi ,C,D)$ , where ${C \subset X, D \subset Y}$ are discrete cross sections and $\psi \colon (C,S_{C}) \to (D,T_{D})$ is a topological conjugacy.

The following result is originally due to Parry and Sullivan in [Reference Parry and SullivanPS75], and is fundamental in the study of flow equivalence; we give a version that is worded differently, more in line with the proof given in [Reference Boyle, Carlsen and EilersBCE17b, Reference Boyle, Carlsen and EilersBCE17a].

More background on flow equivalence can be found in [Reference Boyle, Carlsen and EilersBCE17b, Reference Boyle, Carlsen and EilersBCE17a].

4.3 Flow equivalence of Sturmian, Denjoy, and interval exchange systems

4.3.2 Flow equivalence of Denjoy systems

The following generalization of Theorem 4.5 was proved by Putnam in the context of $C^{*}$ -algebras. The proof in [Reference PutnamPut88, Theorem 6] is stated just for homeomorphisms of the suspensions, but it can be seen from the proof that one can find a homeomorphism respecting orientation of the flow (see also [Reference Aranson, Belitsky and ZhuzhomaABZ96, 6.1.5]).

Theorem 4.7. (Flow equivalence of Denjoy systems)

For $k\in \{0,1\}$ , the Denjoy systems $({\mathcal {D}}_{F_k}, F_k)$ with invariants $(\alpha _k, A_k)$ (satisfying $0\in A_k$ ) are flow equivalent if and only if there exists $M= (m_{ij}) \in \operatorname {\mathrm {GL}}_2(\mathbb {Z})$ such that

(3) $$ \begin{align} \alpha_1 = \frac{m_{11}\cdot\alpha_0+m_{12}}{m_{21}\cdot\alpha_0+m_{22}} \quad \text{and} \quad A_1 \equiv \frac{1}{m_{21}\cdot\alpha_0+m_{22}} \cdot A_0, \end{align} $$

where $\equiv $ denotes equality of subsets of $\mathbb {T}$ up to rotation.

Remark 4.8. (Flow equivalence preserving the cyclic order on fibers)

Similar to the case of Sturmian systems, for $k\in \{0,1\}$ , the Denjoy systems $({\mathcal {D}}_{F_k}, F_k)$ with invariants $(\alpha _k, A_k)$ (satisfying $0\in A_k$ ) are related by a flow equivalence preserving the cyclic order on the fibers if and only if there exists $M= (m_{ij}) \in \operatorname {\mathrm {GL}}_2(\mathbb {Z})$ satisfying the relations in the previous Theorem 4.7 with $\det (M)>0$ and ${1}/({m_{21}\cdot \alpha _0+m_{22}})>0$ .

The orientation reversing homeomorphism of $\mathbb {T}=\mathbb {R}/\mathbb {Z}$ given by $x \mapsto -x$ induces a conjugacy (hence, a flow equivalence) between the Denjoy systems $({\mathcal {D}}_{\alpha , A},F_{\alpha , A})$ and $({\mathcal {D}}_{-\alpha , -A}, F_{-\alpha , -A})$ , which reverses the cyclic order on the fibers.

5.1 2-asymptotic factor maps

Let $(X,S)$ be a system. We say two points $x,x^{\prime } \in X$ are asymptotic, denoted $x\Bumpeq x'$ , when $d(S^{n}(x),S^{n}(x^{\prime })) \to 0$ as $n \to -\infty $ and $n\to +\infty $ . This is an equivalence relation.

Note that for a factor map $\pi \colon (X,S)\to (Y,T)$ , we have $x\Bumpeq x' \implies \pi (x)\Bumpeq \pi (x')$ .

Note that a topological conjugacy is a 2-A factor, since one may choose the set B in the definition to be empty.

Consider a 2-A factor $\pi \colon (X,S) \to (Y,T)$ . Choose a representative $y\in B\bmod {T}$ in every blown-up orbit, and an order on its preimage $\pi ^{-1}(y)=\{x, x'\}$ . For such an asymptotic pair $(x, x')$ , there is a group homomorphism $C(X,\mathbb {Z}) \to \mathbb {Z}$ defined by

$$ \begin{align*} f \mapsto \sum_{n \in \mathbb{Z}}f(S^{n}(x))-f(S^{n}(x')), \end{align*} $$

which only depends on the orbit $y\bmod {T}$ and the order of its preimage $(x, x')\bmod {S}$ . This homomorphism vanishes on coboundaries, so it induces a homomorphism ${\mathcal {I}}_{y} \colon {\mathcal {G}}_{S} \to \mathbb {Z}$ . For every $f \in C(X,\mathbb {Z})$ , there are only a finite number of $y\in B\bmod {T}$ such that ${\mathcal {I}}_y(f)\ne 0$ ; hence, following [Reference Giordano, Putnam and SkauGPS01], we may define the index homomorphism

$$ \begin{align*} {\mathcal{I}} \colon {\mathcal{G}}_{S} \to \bigoplus_{B\bmod{T}} \mathbb{Z} \quad \mathrm{by} \ {\mathcal{I}} = \bigoplus_{y\in B\bmod{T}} {\mathcal{I}}_{y}. \end{align*} $$

Recall that for any factor $\pi \colon (X,S)\to (Y,T)$ between minimal Cantor systems, the map $\pi ^*\colon {\mathcal {G}}_{T}\to {\mathcal {G}}_{S}$ is injective. When $\pi $ is 2-A, the cokernel of $\pi ^*$ is identified with the index map by [Reference Giordano, Putnam and SkauGPS01, Theorem 3.1] as follows.

Theorem 5.5. (Index exact sequence)

Let $\pi \colon (X,S) \to (Y,T)$ be a 2-A factor map between minimal Cantor systems. The index homomorphism ${\mathcal {I}}\colon {\mathcal {G}}_{S} \to \bigoplus _{B\bmod {T}}\mathbb {Z}$ is surjective and its kernel is the image of the induced homomorphism $\pi ^{*}\colon {\mathcal {G}}_{T} \to {\mathcal {G}}_{S}$ . In other words, there is a short exact sequence of abelian groups:

$$ \begin{align*} 0 \to {\mathcal{G}}_{T} \xrightarrow{\pi^*} {\mathcal{G}}_{S} \xrightarrow{{\mathcal{I}}} \bigoplus_{B\bmod{T}}\mathbb{Z} \to 0. \end{align*} $$

5.2 Infinitesimal 2-asymptotic equivalence

Definition 5.6. (infinitesimal)

We say that a 2-A factor $\pi \colon (X,S) \to (Y,T)$ with index homomorphism ${\mathcal {I}}$ is infinitesimal when ${\mathcal {I}}({\mathcal {G}}_{S})/{\mathcal {I}}(\operatorname {\mathrm {Inf}}_{S})$ is torsion, namely

$$ \begin{align*}({\mathcal{I}}({\mathcal{G}}_{S}) / {\mathcal{I}}(\operatorname{\mathrm{Inf}}_{S})) \otimes \mathbb{Q} = 0.\end{align*} $$

Note that if $\pi $ is a topological conjugacy, the associated index map is the zero map and, hence, $\pi $ is an infinitesimal 2-A factor map. Thus, we may define the notion of infinitesimal 2-A equivalence.

The following propositions connect infinitesimal 2-A factors with the rational images of respective states.

Proposition 5.8. (2-AI factors and states)

Consider an infinitesimal 2-asymptotic factor $\pi \colon (X,S) \to (Y,T)$ whose index homomorphism ${\mathcal {I}}$ satisfies $\ker ({\mathcal {I}})\subset \pi ^*({\mathcal {G}}_{T})$ (as it is the case when $(X,S)$ and $(Y,T)$ are minimal Cantor systems by Theorem 5.5). Then, for any state $\tau _X$ of $(X,S)$ , its push-forward state $\tau _Y=\pi _*(\tau _X)$ of $(Y,T)$ satisfies

$$ \begin{align*} \operatorname{\mathrm{im}}(\tau_{X}) \otimes \mathbb{Q} = \operatorname{\mathrm{im}}(\tau_{Y}) \otimes \mathbb{Q}. \end{align*} $$

Proof. Let $[f] \in {\mathcal {G}}_{S}$ . Since ${\mathcal {I}}({\mathcal {G}}_{S})/{\mathcal {I}}(\operatorname {\mathrm {Inf}}_{S})$ is torsion, we may choose $r\in \mathbb {N}_{>0}$ and $g \in \operatorname {\mathrm {Inf}}_{S}$ such that $r{\mathcal {I}}(f) = {\mathcal {I}}(g)$ . Thus, ${\mathcal {I}}(rf - g) = 0$ , so the assumption $\ker {\mathcal {I}} = \operatorname {\mathrm {im}}(\pi ^*)$ ensures that we may choose $h \in C(Y,\mathbb {Z})$ such that $\pi ^{*}(h) = rf - g$ .

Now, consider a state $\tau _X$ of $(X,S)$ and let $\tau _Y=\pi _*(\tau _X)$ be the push-forward state of $(Y,T)$ . By definition of $\operatorname {\mathrm {Inf}}_{S}$ , we have $\tau _X(g)=0$ , thus,

$$ \begin{align*} \tau_{Y}(h) = \pi_{*}(\tau_{X})(h) = \tau_{X}(\pi^{*}(h)) = \tau_{X}(rf - g)=r\tau_X(f), \end{align*} $$

proving that $\operatorname {\mathrm {im}}(\tau _X)\otimes \mathbb {Q} = \operatorname {\mathrm {im}}(\tau _Y)\otimes \mathbb {Q}$ , as desired.

Proof. To show that ${\mathcal {I}}({\mathcal {G}}_{S})/{\mathcal {I}}(\operatorname {\mathrm {Inf}}_{S})$ is torsion, we fix $b\in {\mathcal {I}}({\mathcal {G}}_{S})$ and will construct $f'\in \operatorname {\mathrm {Inf}}_S$ and $d\in \mathbb {N}_{>0}$ such that ${\mathcal {I}}(f')=db$ .

We may choose $f\in {\mathcal {G}}_{S}$ with ${\mathcal {I}}(f)=b$ . By assumption $\operatorname {\mathrm {im}}(\tau _X)\otimes \mathbb {Q}=\operatorname {\mathrm {im}}(\tau _Y)\otimes \mathbb {Q}$ , so there exists $d\in \mathbb {N}_{>0}$ and $g\in {\mathcal {G}}_{T}$ such that $\tau _X(f)=({1}/{d})\tau _Y(g)$ . Let $f'=df- \pi ^*(g)\in {\mathcal {G}}_{S}$ . On the one hand,

$$ \begin{align*}\tau_X(f')=d\tau_X(f)-\tau_X(\pi^*(g))=d\tau_X(f)-\pi_*(\tau_X)(g)=d\tau_X(f)-\tau_Y(g)=0\end{align*} $$

so $f'\in \ker (\tau _X)$ ; hence, $f'\in \operatorname {\mathrm {Inf}}_{S}$ by unique ergodicity of $(X,S)$ . On the other hand, ${\mathcal {I}}(f')=d{\mathcal {I}}(f)$ since $\pi ^*(g)\in \pi ^*({\mathcal {G}}_{T})\subset \ker {\mathcal {I}}$ by assumption.

Let us sum up the previous results in a particular case of interest.

2-AI extensions can also arise from maximal equicontinuous factors of certain substitution subshifts, as in the following example.

5.3 $\mathbb {M}$ -measured equivalence

In this subsection, we fix a subgroup $\mathbb {M}\subset \mathbb {R}$ and generalize the notion of infinitesimal 2-A equivalence to that of $\mathbb {M}$ -measured equivalence.

Note that for any subgroup $\mathbb {M} \subset \mathbb {R}$ , a topological conjugacy is always $\mathbb {M}$ -measured and we may consider the notion of $\mathbb {M}$ -measured 2-A equivalence between uniquely ergodic systems.

Proposition 5.14. ( $\mathbb {M}$ -measured and states)

Consider a 2-asymptotic factor $\pi \colon (X,S) \to (Y,T)$ with index homomorphism ${\mathcal {I}}$ . Fix a state $\tau _X$ of $(X,S)$ and let $\tau _Y=\pi _*(\tau _X)$ be its push-forward state on $(Y,T)$ . Let $\mathbb {M}\subset \mathbb {R}$ be a subgroup. When $\ker ({\mathcal {I}})\subset \pi ^*({\mathcal {G}}_{T})$ , we have the implication:

$$ \begin{align*} \tau_X \text{ is } \mathbb{M}\text{-measured} \implies (\operatorname{\mathrm{im}}(\tau_{X})+\mathbb{M}) \otimes \mathbb{Q} = (\operatorname{\mathrm{im}}(\tau_{Y})+\mathbb{M}) \otimes \mathbb{Q}. \end{align*} $$

When $\ker ({\mathcal {I}})\supset \pi ^*({\mathcal {G}}_{T})$ , we have the implication:

$$ \begin{align*} \tau_X \text{ is } \mathbb{M}-\text{measured} \impliedby (\operatorname{\mathrm{im}}(\tau_{X})+\mathbb{M}) \otimes \mathbb{Q} = (\operatorname{\mathrm{im}}(\tau_{Y})+\mathbb{M}) \otimes \mathbb{Q}. \end{align*} $$

Consequently, when $\ker ({\mathcal {I}})=\operatorname {\mathrm {im}}(\pi ^*)$ , as it is the case when $(X,S)$ and $(Y,T)$ are minimal Cantor systems by Theorem 5.5, we have the equivalence:

$$ \begin{align*} \tau_X \text{ is } \mathbb{M}-\text{measured} \iff (\operatorname{\mathrm{im}}(\tau_{X})+\mathbb{M}) \otimes \mathbb{Q} = (\operatorname{\mathrm{im}}(\tau_{Y})+\mathbb{M}) \otimes \mathbb{Q}. \end{align*} $$

Proof. Suppose that $\ker ({\mathcal {I}})\subset \pi ^*({\mathcal {G}}_{T})$ and that $\pi $ is $\mathbb {M}$ -measured with respect to $\tau _{X}$ . Let $G\subset {\mathcal {G}}_{S}$ be a subgroup such that ${\mathcal {I}}({\mathcal {G}}_{S})/{\mathcal {I}}(G)$ is torsion and $\tau _X(G)\subset \mathbb {M}$ . For every $[f] \in {\mathcal {G}}_{S}$ , we may choose $r\in \mathbb {N}_{>0}$ and $g \in G$ such that $r{\mathcal {I}}(f) = {\mathcal {I}}(g)$ . Thus, ${\mathcal {I}}(rf - g) = 0$ , so we may choose $h \in C(Y,\mathbb {Z})$ such that $\pi ^{*}(h) = rf - g$ . By assumption, $\tau _X(g)\in \mathbb {M}$ , so

$$ \begin{align*}\tau_{Y}(h) = \pi_{*}(\tau_{X})(h) = \tau_{X}(\pi^{*}(h)) = \tau_{X}(rf - g)=r\tau_X(f)-\tau_X(g)\in r\tau_X(f)+\mathbb{M}\end{align*} $$

proving $(\operatorname {\mathrm {im}}(\tau _{X})+\mathbb {M}) \otimes \mathbb {Q} = (\operatorname {\mathrm {im}}(\tau _{Y})+\mathbb {M}) \otimes \mathbb {Q}$ , as desired.

Suppose that $\ker ({\mathcal {I}})\supset \pi ^*({\mathcal {G}}_{T})$ and that $(\operatorname {\mathrm {im}}(\tau _X)+\mathbb {M})\otimes \mathbb {Q} = (\operatorname {\mathrm {im}}(\tau _Y)+\mathbb {M})\otimes \mathbb {Q}$ . Define the subgroup $G=\tau _X^{-1}(\mathbb {M})\subset {\mathcal {G}}_{S}$ and let us show that ${\mathcal {I}}({\mathcal {G}}_{S})/{\mathcal {I}}(G)$ is torsion. Let $b\in {\mathcal {I}}({\mathcal {G}}_{S})$ and choose $f\in {\mathcal {G}}_{S}$ with ${\mathcal {I}}(f)=b$ . Since $(\operatorname {\mathrm {im}}(\tau _X)+\mathbb {M})\otimes \mathbb {Q}=(\operatorname {\mathrm {im}}(\tau _Y)+\mathbb {M})\otimes \mathbb {Q}$ , there exists $d\in \mathbb {N}_{>0}$ , $m\in \mathbb {M}$ , and $g\in {\mathcal {G}}_{T}$ such that $\tau _X(f)=({1}/{d})(\tau _Y(g)+m)$ . Let $f'=df-\pi ^*(g)\in {\mathcal {G}}_{S}$ . On the one hand,

$$ \begin{align*}\tau_X(f')=d\tau_X(f)-\tau_X(\pi^*(g))=d\tau_X(f)-\pi_*(\tau_X)(g)=d\tau_X(f)-\tau_Y(g)=m;\end{align*} $$

thus, $f'\in \tau _X^{-1}(\mathbb {M})=G$ . On the other hand, ${\mathcal {I}}(f')=d{\mathcal {I}}(f)$ since $\pi ^*(g)\in \pi ^*({\mathcal {G}}_{T})\subset \ker {\mathcal {I}}$ by assumption.

5.4 2-A factors of Denjoy and Sturmian systems

In this subsection, we describe all 2-A factors of Denjoy systems (up to topological conjugacy) and characterize which ones are infinitesimal or $\mathbb {M}$ -measured.

The subsets of $\mathbb {R}$ containing $0$ can be added and scaled by real numbers. Hence, the subsets of $\mathbb {T} = \mathbb {R}/\mathbb {Z}$ containing $0$ can be added and multiplied by a real number, by first lifting them along $\mathbb {R} \to \mathbb {R}/\mathbb {Z}$ , then adding them or scaling them, and projecting back down along $\mathbb {R}\to \mathbb {R}/\mathbb {Z}$ . For a subring $\mathbb {K}\subset \mathbb {R}$ and a subset $A \subset \mathbb {T}$ containing $0$ , the $\mathbb {K}$ -submodule of $\mathbb {T}$ generated by A is thus:

$$ \begin{align*} \mathbb{K}{A} &= \{\textstyle{\sum}_i \unicode{x3bb}_i ( \tilde{a_i}+k_i) \bmod{1} \colon \tilde{a_i}\in \tilde{A}\cap [0,1), k_i\in \mathbb{Z}, \unicode{x3bb}_i\in \mathbb{K}\} \\ &= \{\unicode{x3bb}_0+\textstyle{\sum}_i \unicode{x3bb}_i \tilde{a_i} \bmod{1} \colon \tilde{a_i}\in \tilde{A}\cap [0,1), \unicode{x3bb}_i\in \mathbb{K}\} \subset \mathbb{T}. \end{align*} $$

In particular, $\mathbb {K}{A} = \mathbb {K}{(A+\mathbb {K}/\mathbb {Z})}$ .

Fix an irrational $\alpha \in \mathbb {T}$ and an $R_\alpha $ -invariant countable subset $A\subset \mathbb {T}$ containing $0$ . The factor $\pi _{\alpha , A} \colon ({\mathcal {D}}_{\alpha , A}, F_{\alpha , A}) \to (\mathbb {T}, R_{\alpha })$ is 2-asymptotic. Each point $a \in A \subset \mathbb {T}$ has two preimages $a^\pm \in {\mathcal {D}}_{\alpha , A}$ , which form a naturally ordered asymptotic pair for $\pi _{\alpha , A}$ , and all asymptotic pairs arise this way.

Now, for every $R_\alpha $ -invariant subset $A'\subset A$ containing $0$ , there is a factor map $({\mathcal {D}}_{\alpha , A}, F_{\alpha , A}) \to ({\mathcal {D}}_{\alpha ,A}, F_{\alpha , A'})$ obtained by identifying pairs of points $a^-, a^+\in {\mathcal {D}}_{\alpha ,A}$ such that $\pi _{\alpha ,A}(a^-) = \pi _{\alpha , A}(a^+)\in A\setminus A'$ , and these are all its asymptotic pairs.

Now, let us reap from the previous two subsections the corollaries of main interest.

Remark 5.22. (Iterated trees and fibered products of 2-A factors)

There exist 2-A factor maps between zero-dimensional systems with non-trivial fibers lying over points belonging to non-trivial asymptotic classes. We record here an example of such.

Fix an irrational $\alpha \in \mathbb {R}$ and let $(X_{\alpha },\operatorname {\mathrm {\sigma }}_{\alpha })$ be the associated Sturmian subshift with canonical 2-A factor map $\pi \colon X_{\alpha } \to \mathbb {T}$ . Let $(Z,U)$ denote the fiber product of $\pi $ with itself, that is, $Z=\{(x_1,x_2) \in X_{\alpha } \times X_{\alpha } \colon \pi (x_1) = \pi (x_2)\}\subset X_\alpha \times X_\alpha $ with $U= \operatorname {\mathrm {\sigma }} \times \operatorname {\mathrm {\sigma }}$ , so that we have a diagram:

(4)

We claim that $\psi _{1}$ and $\psi _{2}$ are 2-A factor maps. We argue for $\psi _{1}$ . Fix $x_1 \in X_\alpha $ . By construction, $\psi _{1}^{-1}(x_1)= \{(x_1,x_2)\in X_\alpha \times X_\alpha \colon x_2 \in \pi ^{-1}(\pi (x_1))\}$ . Thus, $|{\psi _{1}^{-1}(x_1)}| = |{\pi ^{-1}(\pi (x_1))}|\in \{1,2\}$ is equal to $2$ precisely when $x_1$ belongs to a non-trivial $\operatorname {\mathrm {\sigma }}$ -asymptotic class. Finally, suppose that $\{(x_{1},x_2), (x_{1}, x_2')\}$ is a non-trivial fiber of $\psi _{1}$ . Then, since $\pi (x_{2})=\pi (x_{2}')$ , we must have $x_{2}\Bumpeq x_{2}'$ for $\operatorname {\mathrm {\sigma }}$ ; hence, $(x_{1},x_2)\Bumpeq (x_{1}, x_2')$ for U.

Note that while Z is zero-dimensional, the system $(Z,U)$ is not minimal since the diagonal $\{(x,x) \in X_{\alpha } \times X_{\alpha }\}$ is proper, closed, and invariant. The off-diagonal points are isolated and they form two orbits, which are both asymptotic to the orbits of the non-trivial asymptotic pair in $X_{\alpha }$ . Hence, the factor $Z\to \mathbb {T}$ has non-trivial fibers of cardinal $4$ consisting of asymptotic points.

5.5 2-A factors of interval exchanges and minimal models

Fix our ordered alphabet $\mathcal {A}=\{1,\ldots ,d\}$ on $d\in \mathbb {N}_{\ge 1}$ letters and recall the previous subsections about interval exchange systems $(X,\operatorname {\mathrm {\sigma }})\subset (\mathcal {A}, \operatorname {\mathrm {\sigma }})$ . In particular, if a minimal subshift $(X,\operatorname {\mathrm {\sigma }})\subset (\mathcal {A},\operatorname {\mathrm {\sigma }})$ has complexity $C_X(n)=(d-1)n+1$ , then the IETs with that encoding all have ido, the same irreducible permutation $\pi \colon \mathcal {A}\to \mathcal {A}$ , and their lengths $\unicode{x3bb} \colon \mathcal {A} \to \mathbb {R}_{>0}$ correspond to the invariant measures of $(X, \operatorname {\mathrm {\sigma }})$ .

Let us first describe the 2-A factors of $(X,\operatorname {\mathrm {\sigma }})\subset (\mathcal {A},\operatorname {\mathrm {\sigma }})$ , a minimal IES with complexity $C_X(n)=(d-1)n+1$ and permutation $\pi \colon \mathcal {A}\to \mathcal {A}$ .

The proofs of the following two propositions are left as exercises. They follow from the description in [Reference DahmaniDah19, §2.1] of a minimal model for an IET, combined with the characterization [Reference Ferenczi and ZamboniFZ08] of languages associated to d-IETs with ido (relying on the notion of bispecial words).

Now, assume that $\pi $ is not a cyclic permutation preserving the cyclic order. We say that $\pi \colon \mathcal {A} \to \mathcal {A}$ can be factored at $k\in \mathcal {A}\setminus \{d\}$ when $\pi (k+1)=\pi (k)+1\bmod {d}$ , that is, when T has $\alpha _k$ as a removable discontinuity.

To summarize, any minimal IES $(X,\operatorname {\mathrm {\sigma }})$ with ido has a smallest 2-A factor $(X_{\downarrow }, \operatorname {\mathrm {\sigma }}_{\downarrow })$ that is an IES with ido, which is uniquely determined as a subshift on an ordered alphabet (up to order reversal of the alphabet). Let us note that if an IET has encoding X, then its minimal model will have encoding $(X_{\downarrow }, \sigma _{\downarrow })$ .

6.1 Isogenies of Cantor systems

Proof. The first item is straightforward: the image of $\tau _{\mu } \colon {\mathcal {G}}_{S} \to \mathbb {R}$ is equal to the image of $\tau _{\mu } \colon {\mathcal {G}}_{S^{m}} \to \mathbb {R}$ , and a topological conjugacy $h \colon (X,S^{m}) \to (Y,T^{n})$ that takes $\mu $ to $\nu $ satisfies $\operatorname {\mathrm {im}}(\tau _{\mu }) = \operatorname {\mathrm {im}}(\tau _{\nu })$ .

The second item follows from Proposition 5.8.

For the third item, the flow equivalence $\Psi $ induces an isomorphism $\Psi ^{*} \colon {\mathcal {G}}_{T} \to {\mathcal {G}}_{S}$ . Given $[f] \in {\mathcal {G}}_{S}$ , there exists $[g] \in {\mathcal {G}}_{T}$ such that $\Psi ^{*}([g]) = [f]$ , and hence,

$$ \begin{align*} \tau_{\mu}([f]) = \tau_{\mu}(\Psi^{*}([g])) = \Psi_{*}(\tau_{\mu})([g])=\unicode{x3bb}\tau_{\nu}([g]). \end{align*} $$

It follows that $\operatorname {\mathrm {im}}(\tau _{\mu }) \otimes \mathbb {Q} \subset \unicode{x3bb} \cdot \operatorname {\mathrm {im}}(\tau _{\nu }) \otimes \mathbb {Q} $ . The same argument applied to $\Psi ^{-1}$ implies $\operatorname {\mathrm {im}}(\tau _{\nu }) \otimes \mathbb {Q} \subset \unicode{x3bb} ^{-1} \cdot \operatorname {\mathrm {im}}(\tau _{\mu }) \otimes \mathbb {Q} $ , so $\unicode{x3bb} \cdot \operatorname {\mathrm {im}}(\tau _{\nu }) \otimes \mathbb {Q} \subset \operatorname {\mathrm {im}}(\tau _{\mu }) \otimes \mathbb {Q} $ .

Corollary 6.5. (When isogenies preserve ratios of states)

Suppose $(X,S)$ and $(Y,T)$ are Cantor systems which are totally minimal and totally uniquely ergodic. If they are isogenous, then there exists a positive real $\unicode{x3bb}>0$ such that their unique invariant probability measures $\mu $ and $\nu $ satisfy

$$ \begin{align*} \operatorname{\mathrm{im}}(\tau_\mu)\otimes \mathbb{Q}=\unicode{x3bb} \cdot (\operatorname{\mathrm{im}}(\tau_\nu) \otimes \mathbb{Q}). \end{align*} $$