Continuous Orbit Equivalence

• Continuous orbit equivalence is a topological concept that identifies orbit structures using continuous cocycles, allowing time reparametrization along orbits.
• In symbolic dynamics, this notion appears in one-sided topological Markov shifts via continuous integer-valued time-change functions, distinguishing it from strict conjugacy.
• The framework connects étale groupoids, Cartan-preserving isomorphisms of C*-algebras, and ordered cohomology, offering bridges between dynamical systems and algebraic invariants.

Continuous orbit equivalence is a topological notion of dynamical equivalence in which a homeomorphism identifies orbit structures while allowing a continuous, point-dependent reparametrization of time along orbits. In the general setting of countable discrete groups acting by homeomorphisms on compact Hausdorff spaces, it is encoded by continuous group-valued cocycles; in symbolic dynamics it appears for one-sided topological Markov shifts through continuous integer-valued time-change functions. Across these settings, continuous orbit equivalence is closely tied to étale groupoids, Cartan-preserving isomorphisms of crossed products or Cuntz–Krieger algebras, ordered cohomology, and rigidity phenomena ranging from complete classification theorems to explicit non-conjugate but orbit equivalent examples (Li, 2015, Matsumoto et al., 2013, Jiang, 2018).

1. General definition for topological dynamical systems

Let $G$ and $H$ be countable discrete groups, and let

$G \curvearrowright X,\qquad H \curvearrowright Y$

be continuous actions on compact Hausdorff spaces. In this setting, the actions are continuously orbit equivalent if there exist homeomorphisms

$\varphi:X\to Y,\qquad \psi:Y\to X$

and continuous maps

$c:G\times X\to H,\qquad c':H\times Y\to G$

such that

$$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$

for all $g\in G$, $h\in H$, $x\in X$, and $y\in Y$. Thus $H$0 and $H$1 send orbits to orbits, and the orbit matching is recorded by continuous cocycle maps rather than by a fixed group isomorphism (Jiang, 2018, Li, 2015).

A basic hypothesis in much of the theory is topological freeness. For a group action $H$2, this means that the set of points with trivial stabilizer is dense in $H$3. Under topological freeness, the orbit cocycle maps become genuine cocycles: for example,

$H$4

and for each fixed $H$5, the map $H$6 is a bijection from $H$7 to $H$8. The compatibility identities

$H$9

express invertibility at the cocycle level and underlie the transfer constructions used in homology, cohomology, and groupoid proofs (Jiang, 2018).

For topologically free systems on second countable locally compact Hausdorff spaces, continuous orbit equivalence is equivalent to isomorphism of transformation groupoids and to existence of a $G \curvearrowright X,\qquad H \curvearrowright Y$0-isomorphism

$G \curvearrowright X,\qquad H \curvearrowright Y$1

that sends $G \curvearrowright X,\qquad H \curvearrowright Y$2 onto $G \curvearrowright X,\qquad H \curvearrowright Y$3. This is the topological analogue of the Singer–Feldman–Moore characterization of measurable orbit equivalence via crossed products and Cartan subalgebras (Li, 2015).

2. One-sided topological Markov shifts

A central class of examples is furnished by one-sided topological Markov shifts. For an irreducible, non-permutation $G \curvearrowright X,\qquad H \curvearrowright Y$4-matrix $G \curvearrowright X,\qquad H \curvearrowright Y$5, the shift space

$G \curvearrowright X,\qquad H \curvearrowright Y$6

is compact and zero-dimensional, and the shift map

$G \curvearrowright X,\qquad H \curvearrowright Y$7

defines the one-sided system $G \curvearrowright X,\qquad H \curvearrowright Y$8. For two such systems $G \curvearrowright X,\qquad H \curvearrowright Y$9 and $\varphi:X\to Y,\qquad \psi:Y\to X$0, continuous orbit equivalence means that there exists a homeomorphism $\varphi:X\to Y,\qquad \psi:Y\to X$1 and continuous maps

$\varphi:X\to Y,\qquad \psi:Y\to X$2

such that

$\varphi:X\to Y,\qquad \psi:Y\to X$3

$\varphi:X\to Y,\qquad \psi:Y\to X$4

The associated cocycle functions are $\varphi:X\to Y,\qquad \psi:Y\to X$5 and $\varphi:X\to Y,\qquad \psi:Y\to X$6 (Matsumoto et al., 2013, Matsumoto, 2019).

This notion is strictly weaker than topological conjugacy. In particular, the one-sided shifts defined by

$\varphi:X\to Y,\qquad \psi:Y\to X$7

are continuously orbit equivalent but not conjugate (Matsumoto et al., 2013). At the same time, continuous orbit equivalence is much stronger than a purely set-theoretic orbit correspondence: the time changes $\varphi:X\to Y,\qquad \psi:Y\to X$8 are required to be continuous, and this continuity is precisely what makes groupoid and $\varphi:X\to Y,\qquad \psi:Y\to X$9-algebraic reconstruction possible (Matsumoto et al., 2014).

Several strengthened variants refine the basic relation. Strongly continuous orbit equivalence requires the cocycle to be cohomologous to the constant function $c:G\times X\to H,\qquad c':H\times Y\to G$0; concretely,

$c:G\times X\to H,\qquad c':H\times Y\to G$1

for some $c:G\times X\to H,\qquad c':H\times Y\to G$2, and similarly on the $c:G\times X\to H,\qquad c':H\times Y\to G$3-side. Uniformly continuous orbit equivalence combines continuous orbit equivalence with compatibility on the AF-subgroups of the continuous full groups, and it is equivalent to eventual one-sided conjugacy. The hierarchy

$c:G\times X\to H,\qquad c':H\times Y\to G$4

is strict in general, while uniformly continuous orbit equivalence coincides with eventual one-sided conjugacy (Matsumoto, 2016, Matsumoto, 2014).

The continuous full group $c:G\times X\to H,\qquad c':H\times Y\to G$5 provides a group-theoretic model for these relations. For irreducible, non-permutation matrices, continuous orbit equivalence is equivalent to isomorphism of continuous full groups, and the stronger relations are detected by the AF-subgroup $c:G\times X\to H,\qquad c':H\times Y\to G$6, cocycle subgroups, and coboundary subgroups attached to full-group cohomology (Matsumoto, 2016, Matsumoto, 2020).

3. Groupoids, Cuntz–Krieger algebras, and flow equivalence

To a one-sided shift $c:G\times X\to H,\qquad c':H\times Y\to G$7 one associates the étale groupoid

$c:G\times X\to H,\qquad c':H\times Y\to G$8

Its unit space is $c:G\times X\to H,\qquad c':H\times Y\to G$9, and the Cuntz–Krieger algebra $$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$0 is canonically isomorphic to $$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$1. The canonical diagonal subalgebra $$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$2 is naturally identified with $$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$3 (Matsumoto et al., 2013).

For irreducible one-sided topological Markov shifts, the basic classification theorem states that the following are equivalent: the shifts are continuously orbit equivalent; the groupoids $$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$4 and $$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$5 are isomorphic; there exists an isomorphism $$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$6 with $$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$7; and $$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$8 together with

$$\varphi(gx)=c(g,x)\varphi(x),\qquad \psi(hy)=c'(h,y)\psi(y)$$9

Equivalently, continuous orbit equivalence classes are classified by $g\in G$0, or by $g\in G$1 in Bowen–Franks form (Matsumoto et al., 2013).

The same circle of ideas extends from irreducible topological Markov chains to general shifts of finite type. For one-sided shifts of finite type $g\in G$2 and $g\in G$3, continuous orbit equivalence is equivalent to isomorphism of the associated shift groupoids $g\in G$4 and $g\in G$5. Moreover, continuous orbit equivalence of one-sided shifts implies flow equivalence of the associated two-sided shifts, and two-sided flow equivalence is equivalent to stable groupoid isomorphism,

$g\in G$6

equivalently to Kakutani equivalence or groupoid equivalence (Carlsen et al., 2016).

The operator-algebraic picture also includes circle and gauge actions. For $g\in G$7, one has a circle action $g\in G$8 on $g\in G$9 that is trivial on the diagonal. Continuous orbit equivalence transports these actions through the ordered cohomology map $h\in H$0, while eventual one-sided conjugacy is characterized by diagonal-preserving isomorphisms that exactly intertwine the usual gauge actions. Flow equivalence of the associated two-sided shifts corresponds to stable diagonal-preserving isomorphisms of Cuntz–Krieger algebras compatible with all circle actions trivial on the diagonal, up to cocycles (Matsumoto, 2015).

An analogous picture exists for directed graphs. For graphs in which every cycle has an exit, orbit equivalence in the graph-theoretic sense is equivalent to diagonal-preserving isomorphism of graph $h\in H$1-algebras; the reverse implication holds for arbitrary graphs. The extended Weyl groupoid $h\in H$2 recovers the graph groupoid $h\in H$3 without the topological-principality hypothesis required by Renault’s original construction (Brownlowe et al., 2014).

4. Cohomology, homology, and invariant structures

Continuous orbit equivalence is accompanied by a rich cohomological formalism. For one-sided topological Markov shifts, the ordered cohomology group

$h\in H$4

with positive cone $h\in H$5 plays a central role. Given a continuous orbit equivalence $h\in H$6, Matsumoto defines a homomorphism

$h\in H$7

by an explicit cocycle sum, and this induces an isomorphism

$h\in H$8

The cocycle functions $h\in H$9 and $x\in X$0 represent positive order units, eventually periodic points are preserved, and the induced cohomology isomorphism is the ordered-cohomological shadow of continuous orbit equivalence (Matsumoto et al., 2014, Matsumoto, 2015).

The dynamical consequences include precise relations for periodic orbits and zeta functions. If $x\in X$1 and $x\in X$2 denote the zeta functions of the associated two-sided shifts, then continuous orbit equivalence yields

$x\in X$3

and therefore

$x\in X$4

The same framework shows that the set of shift-invariant regular Borel measures is invariant under continuous orbit equivalence; under the cohomological normalization $x\in X$5 or $x\in X$6, invariant probability measures correspond as well (Matsumoto et al., 2014).

For general topologically free group actions $x\in X$7, Jiang proved that continuous orbit equivalence preserves several Banach-module-valued homology and cohomology theories attached to the action. If $x\in X$8 and $x\in X$9 are the standard coefficient modules built from $y\in Y$0, then for continuously orbit equivalent topologically free actions one has isomorphisms

$y\in Y$1

$y\in Y$2

and

$y\in Y$3

These invariants include the uniformly finite homology and bounded cohomology theories used by Brodzki–Niblo–Nowak–Wright and Monod to characterize topological amenability; because the Johnson class and fundamental class are preserved, topological amenability is invariant under continuous orbit equivalence for topologically free actions (Jiang, 2018).

At the level of full groups, Matsumoto introduced the cohomology group

$y\in Y$4

for the action of the continuous full group $y\in Y$5 on $y\in Y$6, together with cocycle subgroups $y\in Y$7 and coboundary subgroups $y\in Y$8. Continuous orbit equivalence preserves these structures, while strongly continuous orbit equivalence is characterized by preservation of the distinguished cohomology class $y\in Y$9 coming from the full-group cocycle $H$00 (Matsumoto, 2020).

5. Rigidity, strengthened forms, and failures of rigidity

Continuous orbit equivalence is weaker than conjugacy in general. Li constructed topological dynamical systems, including actions of free abelian groups and non-abelian free groups, that are continuously orbit equivalent but not conjugate. At the same time, he proved strong rigidity results: general topological Bernoulli actions are rigid when compared with actions of nilpotent groups, topological Bernoulli actions of duality groups are rigid when compared with actions of solvable groups, and the same is true for certain subshifts of full shifts over finite alphabets (Li, 2015).

For one-sided topological Markov shifts, strengthened forms of continuous orbit equivalence impose progressively tighter control on the time-change cocycles. Strongly continuous orbit equivalence requires the cocycle class of $H$01 to be preserved in ordered cohomology, and uniformly continuous orbit equivalence requires, in addition, compatibility with the AF-subgroups of the continuous full groups. Uniformly continuous orbit equivalence is equivalent to eventual one-sided conjugacy, while strongly continuous orbit equivalence is characterized by diagonal-preserving isomorphisms of Cuntz–Krieger algebras that give cocycle-conjugate gauge actions (Matsumoto, 2016, Matsumoto, 2014).

A major consequence is that strongly continuous orbit equivalence of one-sided topological Markov shifts implies topological conjugacy of the associated two-sided shifts. This places strong continuous orbit equivalence strictly between one-sided conjugacy and ordinary continuous orbit equivalence: it is weaker than one-sided conjugacy, but strong enough to recover the full two-sided dynamics (Matsumoto, 2014). In general shift-space language, strongly continuous orbit equivalence implies two-sided conjugacy (Carlsen et al., 2016).

Rigidity also depends sharply on the acting group. For minimal actions of the infinite dihedral group $H$02 on infinite compact Hausdorff spaces, continuous orbit equivalence implies conjugacy. By contrast, this fails for certain other virtually cyclic groups: if $H$03 is a nontrivial finite group with trivial center, there exist minimal free actions of $H$04 that are continuously orbit equivalent but not conjugate (Jiang, 2021).

A parallel rigidity phenomenon appears for automorphism systems of étale equivalence relations. For topologically free, expansive automorphism actions on compact connected metrizable groups, weak continuous orbit equivalence of the associated automorphism systems is equivalent to conjugacy of the systems and to continuous orbit equivalence of the underlying group actions. If the homoclinic group is dense, these conditions are also equivalent to algebraic conjugacy (Qiang et al., 2021).

6. Entropy, KMS states, and later extensions

The relation between continuous orbit equivalence and entropy is subtle. For continuously orbit equivalent one-sided topological Markov shifts $H$05 and $H$06, Matsumoto related the gauge-action KMS states on the Cuntz–Krieger algebras to the cocycle functions $H$07 and $H$08. Writing $H$09 and $H$10 for the Perron–Frobenius eigenvalues and $H$11 for the unique gauge KMS states, he proved

$H$12

$H$13

If the shifts are strongly continuously orbit equivalent, then $H$14 is cohomologous to $H$15, and the topological entropies coincide (Matsumoto, 2019).

General continuous orbit equivalence, however, does not preserve entropy. A later result shows that for any irreducible non-permutation matrix $H$16, the continuous orbit equivalence class of $H$17 contains one-sided topological Markov shifts whose topological entropy is greater than an arbitrary prescribed positive real number, and also contains shifts whose entropy is less than an arbitrary prescribed positive real number. In particular, topological entropy is not an invariant of continuous orbit equivalence for one-sided topological Markov shifts (Matsumoto et al., 15 Jun 2026).

The framework has also been extended beyond global group actions. For partial dynamical systems, an orbit morphism $H$18 defines a category of partial actions in which isomorphisms are equivalent to continuous orbit equivalences that preserve stabilizers, or equivalently essential stabilizers. When essential stabilizers are torsion-free and abelian, this is further equivalent to a diagonal-preserving isomorphism of the corresponding crossed products (Castro et al., 2024). For semigroup actions by surjective local homeomorphisms, continuous one-sided orbit equivalence is characterized by isomorphism of the associated semi-groupoids, and continuous orbit equivalence is characterized by isomorphism of the transformation groupoids; for totally disconnected spaces this is again equivalent to a diagonal-preserving isomorphism of the reduced groupoid $H$19-algebras (Qiang et al., 2021).

Taken together, these results show that continuous orbit equivalence is best understood as a groupoid-centered equivalence relation. It preserves orbit structure together with continuous cocycle data, often preserves ordered cohomology and amenability-type invariants, and in specific rigid settings collapses to conjugacy. At the same time, it is flexible enough to allow large variations in entropy and one-sided symbolic structure within a single equivalence class.