Contracting Self-Similar Inverse Semigroups

• Contracting self-similar inverse semigroups are inverse semigroups of partial symmetries whose restrictions stabilize via finite nuclei or semi-nuclei.
• They extend classical self-similar group theory to partial actions on topological Markov chains and substitution tiling spaces, revealing new dynamical structures.
• This framework unifies aspects of symbolic dynamics, substitution tilings, and groupoid C*-algebra theory, offering tools for analyzing complexity and simplicity.

Contracting self-similar inverse semigroups are inverse semigroups of partial symmetries for which restrictions along sufficiently long words eventually lie in a finite controlling set, called a nucleus or, in the partial setting, a semi-nucleus. They are formulated most naturally for partial local homeomorphisms on one-sided topological Markov chains and for inverse semigroups of compact open bisections in étale groupoids. The framework extends the classical theory of contracting self-similar groups to genuinely partial actions, and in substitution tiling theory it yields a semigroup-level reconstruction of the Anderson–Putnam complex, the limit solenoid, and the translational hull (Nekrashevych, 5 Sep 2025, Walton et al., 2021).

1. Algebraic and dynamical framework

An inverse semigroup $S$ is a semigroup in which every element has a unique inverse satisfying the Wagner–Preston axioms: for every $s\in S$ there exists a unique $s^{-1}\in S$ such that

$$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$

Its idempotents form a semilattice

$E(S)=\{e\in S:e^2=e\},$

and the natural partial order is

$$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$

This order encodes restriction of partial symmetries, while idempotents encode domains and ranges (Nekrashevych, 5 Sep 2025).

A standard self-similar setting starts with a finite alphabet $X$, a one-sided topological Markov chain $F\subset X^\omega$, and the local inverse maps $S_x(w)=xw$ on cylinders. A set $A$ of local homeomorphisms between clopen subsets of $s\in S$0 is self-similar if for every $s\in S$1 and $s\in S$2,

$s\in S$3

for pairwise distinct $s\in S$4 and $s\in S$5, with the sections $s\in S$6 defined by

$s\in S$7

whenever $s\in S$8 contains $s\in S$9. Equivalently, for all $s^{-1}\in S$0 and $s^{-1}\in S$1, $s^{-1}\in S$2 is either empty or belongs to $s^{-1}\in S$3 (Nekrashevych, 5 Sep 2025).

For inverse semigroups acting by partial maps on a topological Markov chain, self-similarity is expressed by a partial restriction rule. If $s^{-1}\in S$4 acts on $s^{-1}\in S$5 and $s^{-1}\in S$6, then for every $s^{-1}\in S$7 there exist letters $s^{-1}\in S$8 and elements $s^{-1}\in S$9 with disjoint domains such that

$$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$0

and for $$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$1,

$$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$2

where $$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$3 is determined by $$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$4. This is the inverse-semigroup analogue of the classical restriction identity $$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$5, but with partial domains and a domain partition rather than a single globally defined restriction (Walton et al., 2021).

2. Contracting behavior, nuclei, and local contraction

In the groupoid-based formulation, contraction is a finite-state property of restrictions. Let $$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$6 be a shift-invariant étale groupoid over a subshift of finite type $$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$7, with shift functor

$$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$8

The following are equivalent: there exists a compact $$s s^{-1} s = s \quad\text{and}\quad s^{-1} s s^{-1} = s^{-1}.$$9 such that every $E(S)=\{e\in S:e^2=e\},$0 enters $E(S)=\{e\in S:e^2=e\},$1 under some iterate of $E(S)=\{e\in S:e^2=e\},$2; there exists a compact open $E(S)=\{e\in S:e^2=e\},$3 with $E(S)=\{e\in S:e^2=e\},$4 and the same property; and there exists a finite set $E(S)=\{e\in S:e^2=e\},$5 of compact open $E(S)=\{e\in S:e^2=e\},$6-bisections such that for every compact open bisection $E(S)=\{e\in S:e^2=e\},$7 and all $E(S)=\{e\in S:e^2=e\},$8, for $E(S)=\{e\in S:e^2=e\},$9 large enough, the restriction $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$0 belongs to $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$1, including the empty map. The unique smallest such set is the nucleus, and it is stable under sections (Nekrashevych, 5 Sep 2025).

For self-similar inverse semigroups acting on a topological Markov chain, the semigroup adaptation is: $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$2 is contracting if there exists a finite set $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$3 such that for every $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$4 there exists $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$5 with the property that for any word $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$6 of length $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$7 and any $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$8 with $$s\le t \iff s=t e \text{ for some } e\in E(S), \quad\text{equivalently}\quad s=s s^{-1} t.$$9, there are $X$0 and $X$1 satisfying

$X$2

Equivalently,

$X$3

In this partial setting one often works with a finite semi-nucleus $X$4, meaning that every element of $X$5 restricts in one step to elements of $X$6 on appropriate clopen pieces (Walton et al., 2021).

A related but distinct notion is local contractiveness of the standard action on the tight spectrum. For a tree-like inverse semigroup $X$7 such that all tight filters are ultrafilters, Boava and Exel proved that local contractiveness is equivalent to the idempotent criterion

$X$8

with

$X$9

where $F\subset X^\omega$0 means $F\subset X^\omega$1 and there exists $F\subset X^\omega$2 with $F\subset X^\omega$3. In the self-similar cylinder semilattice over $F\subset X^\omega$4, this becomes the requirement that on every nonempty cylinder one can find a subcylinder that is mapped strictly inside itself (Boava et al., 2014). This criterion is closer in spirit to contraction in actions, but it addresses local contractiveness of the tight action rather than the finite-state nucleus property.

3. Limit spaces and asymptotic equivalence

The limit-space theory for contracting self-similar inverse semigroups extends the classical limit space of a self-similar group to partial actions. One starts from the left-infinite path space $F\subset X^\omega$5 of a topological Markov chain. For $F\subset X^\omega$6 and $F\subset X^\omega$7 in $F\subset X^\omega$8, asymptotic equivalence in the semigroup sense is defined by requiring the existence of an infinite tail $F\subset X^\omega$9 and a finite set $S_x(w)=xw$0 such that for all $S_x(w)=xw$1,

$S_x(w)=xw$2

has initial segment $S_x(w)=xw$3. In the contracting case, this can be strengthened so that all $S_x(w)=xw$4 lie in a finite semi-nucleus $S_x(w)=xw$5, and the right tail can be written uniformly as $S_x(w)=xw$6 with $S_x(w)=xw$7 independent of $S_x(w)=xw$8. The limit space is then

$S_x(w)=xw$9

where $A$0 is the transitive closure of the asymptotic equivalence relation, and the left shift induces a map $A$1 (Walton et al., 2021).

The inverse limit

$A$2

is the limit solenoid. In the general groupoid-based framework, contracting groupoids arising from hyperbolic systems admit nuclei whose bi-infinite path equivalence classes produce a solenoid with shift dynamics; in the principal case, the contracting groupoid is equivalent to the stable equivalence groupoid of a hyperbolic system (Nekrashevych, 5 Sep 2025). In the substitution-tiling case, this abstract solenoidal construction becomes concrete: $A$3 is identified with the Anderson–Putnam complex and $A$4 with the translational tiling hull (Walton et al., 2021).

4. Substitution tilings and the canonical inverse-semigroup example

A central family of examples is provided by Kellendonk’s tiling semigroup for a finite local complexity substitution tiling. Fix an FLC, recognisable substitution on a finite prototile set $A$5 that forces the border, possibly after collaring. A doubly pointed patch is a triple $A$6 consisting of a finite patch $A$7 with distinguished tiles $A$8, considered up to translation. The set $A$9 of all such triples, together with $s\in S$00, becomes an inverse semigroup under

$s\in S$01

when the two pointed patches are compatible and the in/out tiles match after translation, and $s\in S$02 otherwise. The inverse is $s\in S$03, the idempotents are $s\in S$04 and $s\in S$05, and the natural order satisfies

$s\in S$06

as doubly pointed patches with the same in/out tiles. Each $s\in S$07 acts as a partial bijection on the discrete hull $s\in S$08 with domain

$s\in S$09

and range $s\in S$10, and these elements correspond to open bisections in the étale translation groupoid (Walton et al., 2021).

Self-similarity is obtained by passing to the substitution graph $s\in S$11, whose vertices are the prototiles and whose edges are supertile extensions $s\in S$12 with $s\in S$13. The associated topological Markov chain $s\in S$14 consists of right-infinite, left-pointing paths, and there is a canonical homeomorphism $s\in S$15. Under this identification, the partial translations coming from doubly pointed patches satisfy the inverse-semigroup restriction rule described above. For recognisable border-forcing substitutions, the tiling semigroup $s\in S$16 is self-similar. It is also contracting: one may take as a finite semi-nucleus the set of doubly pointed star-patches $s\in S$17, namely patches whose tiles share a common point, with either $s\in S$18 or $s\in S$19 and $s\in S$20 adjacent. The contraction mechanism is geometric: under successive restrictions, the distance $s\in S$21 between the in/out tiles shrinks by at least $s\in S$22 per step, where $s\in S$23 is the inflation scale, and finite local complexity leaves only finitely many configurations below a fixed radius. After enough steps, the restrictions are forced into the finite family of star patches (Walton et al., 2021).

This contracting structure controls the limit space. The map $s\in S$24 is defined by intersecting nested subtile supports determined by successive embeddings; its value is a unique point in the support of the range prototile. The semigroup asymptotic equivalence relation corresponds exactly to the gluing relation in the Anderson–Putnam complex $s\in S$25, and therefore

$s\in S$26

Under this identification, the shift $s\in S$27 on $s\in S$28 coincides with the substitution-induced map on $s\in S$29, and the inverse limit $s\in S$30 is homeomorphic to the translational tiling hull $s\in S$31. In the border-forcing Fibonacci example, collaring produces prototiles $s\in S$32 with substituted images

$s\in S$33

and the generators of the semigroup are doubly pointed patches on single tiles or adjacent two-tile patches. The explicit self-similarity rules close on a finite set after enough restriction steps, and the resulting limit space agrees with the Anderson–Putnam complex for the Fibonacci tiling (Walton et al., 2021).

5. Tight groupoids, Steinberg algebras, and simplicity

The inverse-semigroup viewpoint is inseparable from the groupoid of germs and the tight spectrum. For an inverse semigroup $s\in S$34 of local homeomorphisms of a totally disconnected space $s\in S$35, the groupoid of germs has arrows $s\in S$36, with multiplication

$s\in S$37

and inversion

$s\in S$38

For an abstract inverse semigroup with idempotents $s\in S$39, ultrafilters $s\in S$40 form the tight spectrum, and the tight étale groupoid $s\in S$41 is built from germs $s\in S$42 with $s\in S$43. In the ample case, the ultrafilter space is homeomorphic to the unit space, and the groupoid of germs reconstructs the original étale groupoid from the inverse semigroup of compact open bisections. The corresponding Steinberg algebra $s\in S$44 is the $s\in S$45-linear span of indicators of compact open bisections, with convolution

$s\in S$46

and for complex coefficients one obtains the dense $s\in S$47-algebra underlying the groupoid $s\in S$48-algebras (Nekrashevych, 5 Sep 2025).

Contracting self-similar groupoids have particularly strong structural properties. The nucleus determines a Hausdorff criterion: such a groupoid is Hausdorff if and only if, for some Markov partition of the shift, the associated nucleus consists of pairwise disjoint bisections. Contracting groupoids also have polynomial growth of balls and polynomial complexity of Cayley balls, hence are amenable; if $s\in S$49 is a primitive one-sided SFT and the groupoid is contracting and $s\in S$50-invariant, then it is almost finite. These dynamical finiteness properties feed directly into the analysis of the reduced and full groupoid $s\in S$51-algebras and into the structure of topological full groups (Nekrashevych, 5 Sep 2025).

For self-similar groupoids acting on graph path spaces, the associated inverse semigroup $s\in S$52 is generated by paths, path inverses, groupoid elements, and $s\in S$53, and every nonzero element has a canonical form $s\in S$54. The contracted algebra $s\in S$55 has a tight ideal $s\in S$56, and when $s\in S$57 is congruence-free the singular ideal $s\in S$58 is the unique maximal ideal containing $s\in S$59; thus $s\in S$60 is simple exactly when $s\in S$61. In the contracting case, recurrent subgroups inside the nucleus and synchronized vertices of an associated Schreier graph provide an explicit criterion for deciding simplicity. At the $s\in S$62-level, finite closure of isotropy fibers implies that for the tight groupoid $s\in S$63,

$s\in S$64

This extends to contracting self-similar groupoids the coincidence of Steinberg simplicity and reduced $s\in S$65-simplicity already familiar in several special cases (Aakre, 22 Oct 2025).

6. Related theories, misconceptions, and current boundaries

A common misconception is that every inverse-semigroup model of self-similarity presently comes with a developed contracting theory. In the finitely aligned self-similar $s\in S$66-graph framework, the 2024 inverse-semigroup approach does not define a contracting property. Instead, it develops conditions such as pseudo-freeness, locally exhausted strongly fixed sets $s\in S$67, $s\in S$68-cofinality, and $s\in S$69-aperiodicity. These govern Hausdorffness of the tight groupoid, minimality, effectiveness, and simplicity of the associated $s\in S$70-algebra. The paper explicitly presents a possible analogue of contraction in terms of the cocycle $s\in S$71, eventual finiteness of restrictions, and a finite set of “nucleus” paths, but it does not undertake that analysis (Larki, 2024).

The currently established substitution-tiling theory also has clear hypotheses and limits. The tiling constructions use finite local complexity, recognisability, and border forcing in an essential way, with collaring invoked when needed to force the border. Primitivity ensures repetitivity and compactness of the hull in the standard substitution setting. The semigroup paper does not claim uniqueness or minimality of the semi-nucleus, and the choice of finite controlling set can depend on adjacency conventions, such as face-sharing versus mere boundary intersection in higher dimensions. The translational case is the one fully developed; extensions to rigid motions are mentioned, but non-FLC tilings, non-border-forcing substitutions, and questions about how contraction interacts with groupoid hyperbolicity or with $s\in S$72-algebra properties such as pure infiniteness are left open (Walton et al., 2021).

Taken together, these developments show that contracting self-similar inverse semigroups occupy a precise intermediate position between inverse-semigroup models of local symmetry and finite-state dynamics on symbolic or tiling structures. Their finite nuclei or semi-nuclei encode the asymptotic stabilization of restrictions; their limit spaces organize geometric identifications such as the Anderson–Putnam complex; and their tight groupoids mediate the passage to Steinberg algebras, $s\in S$73-algebras, and full groups. This suggests a broad unifying role for contraction across symbolic dynamics, substitution tilings, and self-similar groupoid theory, while also making clear that the notion is not yet uniformly developed in all inverse-semigroup models.