Hom Tree-Shift Dynamics

• Hom tree-shift is a class of tree-indexed symbolic dynamical systems that lifts a one-sided shift space to a regular rooted tree, enforcing branchwise admissibility on every infinite branch.
• Its structure preserves one-dimensional SFT and sofic properties while introducing unique mixing behaviors through complete prefix code variants and branch coupling.
• The framework enables graph-theoretic, entropy, and conjugacy analyses that highlight phenomena like entropy amplification and rigid algorithmic conjugacy recognition.

Hom tree-shift denotes a class of tree-indexed symbolic dynamical systems obtained by lifting a one-sided shift space to a regular rooted tree. If $X\subseteq \mathcal A^{\mathbb Z_+}$, then the associated hom tree-shift is

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$

so every infinite branch of the tree reads an $X$-admissible one-sided sequence (Ban et al., 2021). In the Markov case one writes $X=X_A$ and $\mathcal T_X=\mathcal T_A$, while a graph-theoretic formulation defines a Hom tree-shift $X(G)$ as the set of trees labeled by vertices of a graph $G$ such that each path of the tree is a path of $G$ (Ban et al., 2021). The theory combines one-dimensional symbolic dynamics, tree-shifts, complete-prefix-code mixing, and Markov/graph presentations, and it exhibits phenomena that do not reduce to the one-dimensional case, notably in mixing, entropy, and conjugacy (Ban et al., 2020, Béal et al., 29 Sep 2025).

1. Definition, branchwise admissibility, and local models

A hom tree-shift is defined on the regular rooted tree $\Sigma_k^\ast$, where a labeled tree is a map

$t:\Sigma_k^\ast \to \mathcal{A}.$

A chain is an infinite path

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$0

and the projection of a tree along a chain is the one-sided sequence determined by the labels on that path. The defining condition for $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$1 is therefore branchwise: every root-to-infinity branch must belong to $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$2 (Ban et al., 2021).

The same source records an alternative “axial” version,

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$3

and states that $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$4 and $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$5 coincide when $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$6 is a Markov shift $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$7. This distinction matters because branchwise admissibility is stronger than merely requiring admissibility along coordinate rays; for the even shift, the axial tree-shift can allow a linking pattern that $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$8 does not (Ban et al., 2021).

A graph-theoretic formulation appears in later work: for a simple graph $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$9 with loops allowed, the Hom tree shift $X$0 is the set of trees $X$1 labeled on $X$2 such that each path of $X$3 is a path of $X$4. Equivalently, for every node $X$5 and child index $X$6,

$X$7

This realizes Hom tree-shifts as nearest-neighbor tree-shifts of finite type. The forbidden patterns are precisely the height-$X$8 blocks $X$9 for which $X=X_A$0 is not an edge of $X=X_A$1 for at least one $X=X_A$2 (Béal et al., 29 Sep 2025).

2. SFT, soficity, and finite presentations

A central structural result is that the hom-tree construction preserves the basic one-dimensional symbolic classes. If $X=X_A$3 is a shift space, then

$X=X_A$4

and if $X=X_A$5 is a shift space, then

$X=X_A$6

(Ban et al., 2021).

In the SFT case, the construction is explicit. If $X=X_A$7 is an SFT with a finite forbidden list in $X=X_A$8, then $X=X_A$9 is described by a finite forbidden list of tree blocks on $\mathcal T_X=\mathcal T_A$0. Conversely, if $\mathcal T_X=\mathcal T_A$1 is a tree-SFT with forbidden blocks on some finite initial subtree $\mathcal T_X=\mathcal T_A$2, then one recovers a finite forbidden set in $\mathcal T_X=\mathcal T_A$3 defining $\mathcal T_X=\mathcal T_A$4. For the golden mean shift $\mathcal T_X=\mathcal T_A$5 with forbidden word $\mathcal T_X=\mathcal T_A$6, the corresponding forbidden tree patterns are

$\mathcal T_X=\mathcal T_A$7

which directly encode the branchwise exclusion of consecutive $\mathcal T_X=\mathcal T_A$8's (Ban et al., 2021).

The sofic case is subtler. A useful characterization is that a tree-shift is sofic if and only if there exist a Markov tree-shift $\mathcal T_X=\mathcal T_A$9 and a symbol map $X(G)$0 such that the image of $X(G)$1 is $X(G)$2. The literature also emphasizes that a cover of $X(G)$3 need not arise as $X(G)$4 for a one-dimensional cover $X(G)$5. The even shift is the standard example: $X(G)$6 requires an indirect two-step graph construction rather than the naive lift of the original cover (Ban et al., 2021).

These results place hom tree-shifts inside the usual finite-type/sofic hierarchy, but with the branchwise lift imposing nontrivial compatibility conditions across all branches simultaneously. A plausible implication is that hom tree-shifts should be viewed neither as merely one-dimensional shifts replicated on a tree nor as arbitrary tree-SFTs; they occupy a constrained intermediate position.

3. Mixing theory and comparison with the base shift

The mixing theory of hom tree-shifts is formulated in terms of both ordinary tree-shift notions and complete prefix code (CPC) variants. The standard properties are irreducibility (IR), topological mixing (TM), block gluing (BG), and strong irreducibility (SI). Their CPC analogues replace large-distance separation by placement along a complete prefix code, and uniform CPC versions require the code to be of the form $X(G)$7 (Ban et al., 2021).

For arbitrary tree-shifts, the general implication pattern includes: $X(G)$8 For hom tree-shifts $X(G)$9, several distinctions collapse. The paper proves: $G$0 Thus, for $G$1, uniform CPC block gluing becomes the key intermediary and forces several other tree-mixing properties (Ban et al., 2021).

The relation between $G$2 and $G$3 is asymmetric in general. If $G$4 is TM, then $G$5 is mixing; if $G$6 is IR, then $G$7 is transitive. The converses fail. The even shift $G$8 is mixing, but $G$9 is not even CPC irreducible. Likewise, for the one-sided bounded density shift $G$0 with

$G$1

the base shift is mixing, while $G$2 is TM but not CPC UBG, hence neither UBG nor BG (Ban et al., 2021).

The Markov case is substantially tighter. If $G$3 is the one-sided Markov shift induced by an adjacency matrix $G$4, then

$G$5

The same source notes the combined consequence that, for Markov shifts,

$G$6

(Ban et al., 2021).

A common misconception is that branchwise lifting preserves one-dimensional mixing behavior automatically. The counterexamples above show that branching can create global incompatibilities not visible on any single branch constraint.

4. Entropy, higher-block invariance, and reducible phenomena

For a one-sided shift $G$7, the entropy is

$G$8

whereas for a tree-shift $G$9,

$\Sigma_k^\ast$0

Petersen and Salama showed that the tree entropy exists and satisfies

$\Sigma_k^\ast$1

for any $\Sigma_k^\ast$2 (Ban et al., 2020).

In the irreducible Markov case $\Sigma_k^\ast$3, if

$\Sigma_k^\ast$4

then

$\Sigma_k^\ast$5

Thus equality occurs exactly when all rows of the adjacency matrix have the same sum. The same paper proves that entropy is preserved under higher block presentations of the base shift: $\Sigma_k^\ast$6 This is notable because topological entropy is not a conjugacy invariant for tree-shifts in general, but it remains invariant for hom tree higher block shifts (Ban et al., 2020).

Tree entropy also exhibits specifically tree-dynamical effects. For a binary essential matrix $\Sigma_k^\ast$7, if the maximal row sum $\Sigma_k^\ast$8, then $\Sigma_k^\ast$9. If $t:\Sigma_k^\ast \to \mathcal{A}.$0, then

$t:\Sigma_k^\ast \to \mathcal{A}.$1

This yields a gap phenomenon: the set of topological entropies of hom tree-shifts of finite type is not dense (Ban et al., 2020).

Reducibility produces a second sharp contrast with one-dimensional symbolic dynamics. If $t:\Sigma_k^\ast \to \mathcal{A}.$2 is reducible with irreducible components $t:\Sigma_k^\ast \to \mathcal{A}.$3, then

$t:\Sigma_k^\ast \to \mathcal{A}.$4

but equality need not hold. Explicit examples show that a reducible hom tree-shift can have strictly larger entropy than every irreducible component. A refined analysis for reducible block upper-triangular families $t:\Sigma_k^\ast \to \mathcal{A}.$5 shows that the classical formula $t:\Sigma_k^\ast \to \mathcal{A}.$6 fails in general, and gives exact criteria for when $t:\Sigma_k^\ast \to \mathcal{A}.$7 and when $t:\Sigma_k^\ast \to \mathcal{A}.$8 (Ban et al., 2020, Ban et al., 2021).

This suggests that entropy on hom tree-shifts is governed not only by branchwise admissibility but also by how branching couples admissible extensions across subtrees. The reducible case makes that coupling explicit.

5. Markov hom tree-shifts, large deviations, and Hausdorff dimension

A probabilistic formulation treats a hom tree-shift as the support of a tree-indexed Markov chain. On the rooted $t:\Sigma_k^\ast \to \mathcal{A}.$9-tree

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$00

with transition matrix $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$01, the associated Markov hom tree-shift is

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$02

When $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$03, this reduces to the usual one-sided Markov subshift (Ban et al., 2024).

The same framework yields a tree analogue of Cramér’s theorem. For a positive weight matrix $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$04, the empirical average

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$05

need not converge along the full sequence in the irreducible periodic case. If the distinguished state has period $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$06, the correct objects are the subsequences $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$07, and each of these satisfies a large deviation principle with rate function

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$08

The corresponding almost-sure limit theorem is likewise periodic rather than full-sequence (Ban et al., 2024).

The Hausdorff dimension of $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$09, with respect to the metric

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$10

is given, for irreducible $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$11 of period $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$12, by a nonlinear variational formula involving the transfer operator $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$13: $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$14 In the primitive case $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$15, this reduces to

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$16

More generally,

$$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$17

and equality holds for irreducible $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$18 if and only if $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$19 has uniform row sums (Ban et al., 2024).

These results align with the entropy criterion for equality $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$20 when all row sums are constant. A plausible implication is that uniform row-sum structure is the combinatorial regime in which branching stops amplifying the underlying one-dimensional complexity.

6. Conjugacy, decidability, and related terminology

A graph-defined Hom tree-shift $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$21 is a special tree-SFT, but recent work gives an intrinsic recognition criterion among arbitrary tree-shifts of finite type. The main tool is the edge tree automaton. A tree-shift accepted by a regular edge tree automaton is conjugate to a Hom tree-shift, and a tree-shift of finite type is conjugate to an edge tree shift. The decisive criterion is: $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$22 The resulting decision procedure is explicit: convert the finite-type tree-shift to a trim edge tree automaton, compute its total amalgamation by repeated merging, and check regularity. The paper states that this can be done in polynomial space and time (Béal et al., 29 Sep 2025).

Conjugacy within the Hom class is rigid. Two Hom tree shifts $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$23 and $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$24 are conjugate if and only if $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$25 and $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$26 are isomorphic as undirected graphs. The directed analogue replaces regularity by symmetry: a tree-shift is conjugate to a directed Hom tree-shift if and only if its total amalgamation is symmetric (Béal et al., 29 Sep 2025).

This line of work should be distinguished from the separate “HOM-problem” on tree homomorphisms. There, HOM asks whether the image $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$27 of a regular tree language under a tree homomorphism is again regular, and weighted variants study regularity of homomorphic image series under restrictions such as tetris-free homomorphisms and $$\mathcal T_X:=\{t \in \mathcal{A}^{\Sigma_k^\ast}: \pi_{\mathbf{s}}(t) \in X \text{ for every chain } \mathbf{s}\},$$28-unambiguity or field-valued weights (Nász, 2023, Nász, 2023). That terminology concerns images under tree homomorphisms, not hom tree-shifts in the symbolic-dynamical sense.

Taken together, the literature presents hom tree-shifts as branchwise lifts of one-sided symbolic constraints whose Markov instances admit graph, automaton, entropy, and dimension theories. Their defining feature is simple—every branch must lie in a prescribed one-sided system—but their global behavior is not: mixing notions collapse in distinctive ways, entropy can exceed all irreducible components in reducible cases, and conjugacy to a Hom tree-shift is algorithmically recognizable yet highly rigid once attained.