Restricted Bratteli Diagrams

• RBD is a family of Bratteli diagrams subject to various restrictions on vertices, edges, orders, or incidence matrices, establishing well-structured frameworks.
• These restrictions facilitate rigorous analysis of invariant measures, ergodicity, and dynamical rigidity by controlling combinatorial and algebraic features.
• RBD frameworks span categorical, substitutional, and representation-theoretic formulations, offering practical insights for both theoretical and applied dynamical systems.

Restricted Bratteli Diagrams (RBD) is not a standard term with a single fixed definition in the Bratteli-diagram literature. Across the most relevant papers, it denotes either subdiagrams obtained by restricting vertices or edges levelwise, or classes of Bratteli diagrams constrained by additional combinatorial, categorical, dynamical, geometric, or representation-theoretic conditions. In that sense, RBD refers to a family of restriction mechanisms rather than to one canonical object. The common theme is that restriction is implemented through operations such as telescoping, choice of substructure, prescribed local branching rules, constrained incidence matrices, and diagram maps encoding factors or invariant measures (Adamska et al., 2015, Amini et al., 2015).

1. Terminological scope and core definitions

A Bratteli diagram in the classical graph-theoretic sense consists of a vertex set $V$ and an edge set $E$ with decompositions

$$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$

where each $V_n$ is finite and non-empty, $V_0$ has exactly one element, each $E_n$ is finite and non-empty, and source and range maps satisfy

$$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$

Every vertex emits at least one edge, and every non-root vertex receives at least one edge. Path sets between levels are written

$E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$

and, once vertices in each level are ordered, the edge sets carry multiplicity matrices $M(E_n)$. An ordered Bratteli diagram is such a diagram equipped with a local order on incoming edges: two edges are comparable if and only if they have the same range. Properly ordered diagrams are those with unique infinite maximal and minimal paths, while simple diagrams are those for which some telescoping has strictly positive multiplicity matrices at every level (Amini et al., 2015).

A second, matrix-based formulation writes a Bratteli diagram as $B=(V,E)$ with $E$0 a sequence of positive integer column vectors and $E$1 a sequence of embedding matrices from $E$2 to $E$3. This is the formulation used in the categorical treatment of AF algebras and their Bratteli diagrams (Amini et al., 2014).

The term “restricted” is therefore best understood as referring to extra conditions imposed on one or more of the following: the allowed vertices and edges, the incidence matrices, the admissible orderings, the local branching rules, or the morphisms between diagrams.

Restriction mechanism	Typical form	Representative sources
Subdiagram restriction	Vertex or edge subdiagram; tail-saturation	(Adamska et al., 2015, Bezuglyi et al., 2024)
Level restriction	Telescoping to a cofinal subsequence	(Amini et al., 2015, Amini et al., 2014)
Local branching restriction	$E$4; $E$5; $E$6	(Gaetz, 2018, delMas et al., 2012, Kieffer, 2016)
Order restriction	Perfect orderings; random orders; prescribed extreme paths	(Bezuglyi et al., 2012, Janssen et al., 2014)
Countable-level restriction	Pascal unions; horizontal stationarity; odometer families	(Bezuglyi et al., 2024, Bezuglyi et al., 2024, Bezuglyi et al., 2024)

2. Subdiagrams, thinness, and extension of invariant measures

The most direct mathematical realization of an RBD is a subdiagram. In the standard finite-level setting, a vertex subdiagram $E$7 is determined by proper nonempty subsets $E$8, and retains exactly those edges whose source and range both lie in the chosen subsets. An edge subdiagram keeps all vertices but replaces the incidence matrices $E$9 by smaller matrices $$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$0. For any subdiagram, the natural ambient domain is its tail-saturation

$$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$1

and an ergodic probability measure $$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$2 on $$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$3 extends canonically by tail invariance to a measure $$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$4 on $$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$5. For vertex subdiagrams,

$$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$6

while for edge subdiagrams the analogous formula runs over all vertices in $$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$7. The exact finiteness criteria are expressed by summability of the mass entering the subdiagram from deleted vertices, or by the mass carried by deleted edges (Adamska et al., 2015).

The same paper gives a sharp positivity criterion for the path space of a restricted diagram inside a simple ambient diagram. For a vertex subdiagram $$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$8 with heights $$V = V_0 \sqcup V_1 \sqcup V_2 \sqcup \cdots,\qquad E = E_1 \sqcup E_2 \sqcup E_3 \sqcup \cdots,$$9 inside the subdiagram and ambient heights $V_n$0, one has

$V_n$1

Thus a restricted path space has zero ambient measure exactly when the proportion of ambient paths that stay inside the restriction tends uniformly to zero. The paper calls such a subdiagram thin in effect, and proves that if $V_n$2 is thin, then every probability invariant measure on $V_n$3 has infinite extension to $V_n$4 (Adamska et al., 2015).

In finite-rank theory, this subdiagram perspective becomes structural rather than auxiliary. Every ergodic invariant measure on a finite-rank Bratteli diagram is obtained by extension from a finite ergodic measure on a simple subdiagram, and after telescoping the supports of distinct ergodic measures can be separated into disjoint vertical subsystems. Exact finite rank further imposes a uniform lower bound on tower measures and implies unique ergodicity (Bezuglyi et al., 2010). A later finite-rank analysis reformulates the same idea in simplex language: if a rank-$V_n$5 diagram has exactly $V_n$6 ergodic invariant probability measures, then after telescoping there are $V_n$7 pairwise disjoint subdiagrams $V_n$8 supporting those ergodic directions, together with a residual part that contributes only non-extreme convex combinations (Bezuglyi et al., 2017).

3. Ordered, categorical, and substitutional formulations

Ordered Bratteli diagrams form a category in which isomorphism coincides with equivalence in the sense of Herman, Putnam, and Skau. In that category, the natural correspondence between Cantor minimal systems and simple properly ordered Bratteli diagrams is an equivalence of categories, and factor maps between Cantor minimal systems are modeled by premorphisms between the corresponding ordered diagrams. Telescoping is central throughout: it replaces the original level sequence by a cofinal subsequence and replaces edges by path sets between the new levels, while preserving the underlying Bratteli–Vershik dynamics up to equivalence (Amini et al., 2015).

For ordinary, unordered Bratteli diagrams, a parallel categorical framework is built from premorphisms $V_n$9 satisfying

$V_0$0

with morphisms defined as equivalence classes of such premorphisms. In this category, isomorphism coincides with Bratteli’s original notion of equivalence, and the functor from AF algebras to Bratteli diagrams is full and a strong classification functor. The matrix identity

$V_0$1

characterizes the unital case (Amini et al., 2014).

A substitution-theoretic application shows why the language of restriction cannot be separated from the chosen equivalence relation. Telescope equivalence of stationary Bratteli diagrams preserves primitivity, simplicity, and pure aperiodicity in the irrational Perron–Frobenius sense, but fails to preserve rank, number of letters, and the full order data of substitution words. Ordered telescope equivalence is strictly finer, because it remembers the order of occurrences of letters through edge orders, but it is still not a complete invariant (Gawlak et al., 2021). This suggests that fixed-rank or fixed-alphabet subclasses are natural at the level of presentation, yet not stable under telescope equivalence.

4. Algebraic, representation-theoretic, and regular branching restrictions

One major restricted class arises from towers of finite groups. If

$V_0$2

is a tower of groups, its representation-theoretic Bratteli diagram has vertices $V_0$3, and edge multiplicities given by induction–restriction multiplicities. The restriction becomes rigid when the associated up and down operators satisfy

$V_0$4

equivalently

$V_0$5

Such towers are $V_0$6-dual towers of groups. For $V_0$7 or $V_0$8 prime, the classification theorem states that the only possibilities are the wreath products $V_0$9, and the only Bratteli diagram is $E_n$0, the $E_n$1-fold product of Young’s lattice (Gaetz, 2018).

A second explicitly restricted branching graph is the rook-Brauer Bratteli diagram. Its level-$E_n$2 vertices are the partitions

$E_n$3

and there is an edge from $E_n$4 to $E_n$5 if and only if one of three local moves occurs: $E_n$6 This “stay / add one box / remove one box” rule is the precise restricted branching structure. In the semisimple case, restriction in the tower of rook-Brauer algebras follows exactly this graph: $E_n$7 and irreducible modules are constructed on path bases indexed by these restricted paths (delMas et al., 2012).

An information-theoretic version of restriction is provided by $E_n$8-regular Bratteli diagrams. Here every non-root vertex has exactly $E_n$9 incoming edges, the diagram is isomorphic to a $$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$0-canonical recursive block decomposition, and one can define a Vershik transformation acting like $$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$1-adic addition. On this regular class, the paper proves an ergodic decomposition theorem, an entropy-rate decomposition theorem, a Shannon–McMillan–Breiman theorem, and lossless and lossy source coding theorems. In this setting, the operationally relevant restricted subclass is the class of $$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$2-regular diagrams (Kieffer, 2016).

5. Perfect orderings, random orders, and Vershik obstructions

Restriction may be imposed on orderings rather than on vertices or edges. For finite-rank ordered diagrams, a central problem is whether the partial successor map extends to a continuous Vershik map. Necessary and sufficient conditions for perfectness can be expressed in several equivalent ways: through the language of ordered finite paths, through successor and predecessor correspondences between maximal and minimal paths, and through the skeleton/associated-graph formalism. In the finite-rank case, the existence of perfect orderings with a prescribed number of extreme paths strongly constrains the incidence matrices. In particular, if almost all orderings have $$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$3 maximal and minimal paths and $$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$4 exceeds the number of minimal components, then almost all orderings are imperfect (Bezuglyi et al., 2012).

A different restriction concerns random orders on a highly symmetric infinite-rank class: simple Bratteli diagrams with exactly one edge connecting any two vertices in consecutive levels, so

$$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$5

In this model, a sharp dichotomy holds. If

$$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$6

then almost surely there is a unique maximal path. If

$$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$7

then almost surely there are uncountably many maximal paths. Combined with the general criterion that if random orders typically have more than one maximal and minimal path then almost all such orders are imperfect, this yields a growth-based obstruction to continuous Vershik maps. The paper also proves that, for a large family of impartial, superquadratic, exponentially bounded infinite-rank diagrams, random orders almost surely have infinitely many maximal paths and hence do not admit a continuous Vershik map (Janssen et al., 2014).

These results make clear that order restriction and growth restriction interact in a highly nontrivial way. They also show that telescoping may preserve ordered-diagram equivalence while changing random-order maximal-path behavior.

6. Generalized, regular, and geometric extensions of the restricted viewpoint

Generalized Bratteli diagrams, with countably infinite vertex sets on each level and finite incoming degree, enlarge the scope of restriction while preserving a usable inverse-limit formalism. In this setting, subdiagrams remain the principal restricted objects. A central extension formula for a vertex subdiagram $$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$8 determined by $$s(E_n)\subseteq V_{n-1},\qquad r(E_n)\subseteq V_n.$$9 is

$E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$0

which is finite exactly when the displayed series converges. In reducible generalized diagrams built from infinitely many standard subdiagrams, especially odometers, all ergodic probability measures may be obtained as finite extensions from those restricted components. At the same time, the classical finite-rank principle “distinguished eigenvalues correspond to ergodic probability measures” can fail: an eigenvalue/eigenvector may generate only an infinite $E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$1-finite invariant measure (Bezuglyi et al., 2024).

The inverse-limit method has since been pushed further for generalized diagrams. Probability tail invariant measures are identified with inverse limits of infinite-dimensional simplices, and the extreme points arise from limit row vectors of products of stochastic incidence matrices. This framework is applied to the infinite Pascal graph, to generalized diagrams formed by a countable set of odometers, and to reducible generalized diagrams with uncountably many ergodic probability measures. The same paper also treats vertex and edge subdiagrams and their measure extensions by tail invariance (Bezuglyi et al., 2024).

A particularly transparent recent model is the stationary generalized diagram formed as a union of countably many classical Pascal-Bratteli diagrams. For each $E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$2, the path space $E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$3 of the vertex subdiagram $E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$4 determined by

$E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$5

is the path space of a classical Pascal diagram. After normalization, the measures

$E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$6

are exactly all ergodic probability invariant measures on the ambient generalized diagram. The same paper proves a support-approximation theorem: for every $E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$7 and every $E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$8, there exists a subdiagram $E_{k,\ell}=E_{k+1}\circ\cdots\circ E_\ell,$9 with path space $M(E_n)$0 such that

$M(E_n)$1

This is a direct measure-theoretic realization of restriction by subdiagram (Bezuglyi et al., 2024).

Horizontal stationarity gives another rigid generalized subclass. Here each level is identified with $M(E_n)$2, and the incidence numbers satisfy

$M(E_n)$3

so every incidence matrix is an infinite banded Toeplitz matrix. These diagrams belong to both $M(E_n)$4 and $M(E_n)$5, support a Fourier/convolution reformulation of the invariant-measure equations, and admit an explicit odometer-extension criterion: $M(E_n)$6 For a tridiagonal class $M(E_n)$7, all ergodic tail invariant probability measures are exactly the normalized extensions of the vertical odometer measures (Bezuglyi et al., 2024).

Finally, a geometric version of restriction appears in weighted, fully ordered, bi-infinite Bratteli diagrams. Such a diagram determines a flat surface assembled from rectangles whose widths and heights are given by the positive and negative weights. The shift on the space of bi-infinite diagrams acts as a renormalization dynamics, and if the shift orbit has an accumulation point whose positive part is minimal, then the positive tail equivalence relation, any Bratteli–Vershik transformation defined on it, and the associated vertical translation flow all admit a unique invariant probability measure. This is a Masur-type unique-ergodicity criterion for a restricted weighted and ordered class of diagrams (Treviño, 2016).

Taken together, these developments show that “Restricted Bratteli Diagrams” is best understood as an umbrella concept for diagram classes in which restriction is mathematically explicit: subdiagrams, telescoped systems, regular indegree models, prescribed local branching graphs, constrained orderings, Toeplitz-type generalized diagrams, and weighted bi-infinite renormalization models. The unifying principle is not a single definition, but a repeated strategy: impose structure, identify the resulting invariant-measure or dynamical rigidity, and compare the restricted object to its ambient Bratteli, Cantor, algebraic, or geometric system.