Entropy of Group Endomorphisms

• Entropy of Group Endomorphisms is a measure of the exponential growth of finite subsets under repeated endomorphism action.
• The theory employs the Addition Theorem to decompose entropy across invariant subgroups, applicable in both abelian and torsion nilpotent groups.
• Rigidity features such as the integer–logarithm dichotomy and combinatorial string invariants provide deep insights into the structural dynamics of groups.

An endomorphism entropy theory for groups quantifies the exponential complexity with which iterates of an endomorphism expand finite subsets or subgroups. For group endomorphisms, particularly in abelian and locally finite settings, algebraic entropy and its variants reflect the dynamical growth rate of orbits and provide deep structural insights. The theory features a precise definition of entropy, an Addition Theorem for decomposing entropy across invariant subgroups and quotients, connections to combinatorial invariants such as string numbers, and rigidity properties in various group-theoretic classes. Recent advances have significantly extended these results to nonabelian (e.g., nilpotent) cases, uncovering integer–logarithm dichotomies and structural reduction principles.

1. Definition of Algebraic Entropy for Group Endomorphisms

Given a group $G$ (not necessarily abelian), consider an endomorphism $\varphi\in\mathrm{End}(G)$. For a finite subset or subgroup $F\leq G$ and $n\in\mathbb N_+$, define the $n$-step $\varphi$-trajectory

$$T_n(\varphi,F) := F \cdot \varphi(F) \cdot \cdots \cdot \varphi^{n-1}(F).$$

In the abelian case, sum replaces product. Fekete’s lemma implies that the sequence $a_n = \log|T_n(\varphi,F)|$ is subadditive, ensuring existence of the limit

$$H(\varphi, F) := \lim_{n\to\infty} \frac{1}{n}\log|T_n(\varphi,F)|.$$

The algebraic entropy of $\varphi$ is

$$h(\varphi) := \sup\{H(\varphi, F) : F\le G,\ F\ \text{finite}\} \in [0,\infty].$$

In the broader nonabelian locally finite context—where every finitely generated subgroup is finite—the same definition applies and the limit exists for each finite subgroup $F$ (Shlossberg, 14 Jan 2026). This notion measures the exponential growth rate of the image of $F$ under repeated application of $\varphi$.

Core properties derived from this definition in the abelian case, and in specific nonabelian regimes, include:

• Conjugation invariance: $h(\varphi)$ is preserved under conjugation in the automorphism group of $G$.
• Logarithmic law: $h(\varphi^{k}) = |k|h(\varphi)$ for automorphisms, $h(\varphi^{k}) = k h(\varphi)$ in general.
• Continuity: Under direct systems of $\varphi$-invariant (sub)groups, entropy is the supremum of the entropies on subsystems.
• Monotonicity: $$h(\varphi)\ge \max\{h(\varphi|_H),\, h(\overline\varphi)\}$$ for $\varphi$-invariant $H\leq G$ and quotient map $\overline\varphi$.

2. The Addition Theorem and Its Extensions

A central structural feature is the Addition Theorem (AT), which asserts additivity of entropy across invariant normal subgroups. For $G$ abelian or a torsion nilpotent group and $H\triangleleft G$ normal, $\varphi(H)\le H$, with induced map $\overline\varphi$ on $G/H$, the theorem states:

$h(\varphi) = h(\varphi|_H) + h(\overline\varphi).$

For torsion nilpotent $G$ of arbitrary class and any $\varphi\in\mathrm{End}(G)$, this addition theorem now holds for all $\varphi$-invariant normal subgroups. Proof proceeds by induction on nilpotency class, reduction to central extensions, and subgroup counting, supplemented by glue lemmas to handle successive lower central series steps (Shlossberg, 14 Jan 2026).

Applications include additivity along upper and lower central series, with particular consequences for $\omega$-hypercentral and locally finite groups. In varieties of locally finite groups, AT holds generally if and only if it holds for groups generated by bounded sets (i.e., in some $G[n!]$ with $n!$ annihilating all generators), allowing substantial reduction in verifying the theorem (Shlossberg, 14 Jan 2026).

3. Rigidity and Integer-Logarithm Dichotomy

A distinctive phenomenon in the entropy of torsion nilpotent group endomorphisms is the integer–logarithm dichotomy. Specifically, the following holds:

If $G$ is a torsion nilpotent group and $\varphi\in\mathrm{End}(G)$, then either $h(\varphi) = \infty$ or else $h(\varphi) = \log(\alpha)$ for some $\alpha\in\mathbb N$.

This result arises via structural induction (using the addition theorem) and reflects that, unlike in the torsion-free or general abelian case, no intermediate values (such as irrational or transcendental logarithms) occur (Shlossberg, 14 Jan 2026). Thus, either the entropy measures unbounded exponential complexity or, if finite, corresponds to the log of a natural number, tightly constraining the possible dynamical behaviors.

This dichotomy extends to locally finite $\omega$-hypercentral groups for automorphisms and to general varieties of locally finite groups under the aforementioned reduction (Shlossberg, 14 Jan 2026).

4. Combinatorial and Set-Theoretic Invariants: Strings

Strings capture essential combinatorial aspects of the action of endomorphisms and play a critical role in entropy calculations, especially for shift systems and nontrivial dynamics. For a self-map $X:T\to T$, a string is a sequence $S=\{x_n\}$ with $X(x_{n}) = x_{n-1}$ and all $x_n$ distinct. Several string numbers are defined:

• $s(X)$: supremal number of disjoint strings,
• $ns(X)$: for non-singular strings (not quasi-periodic),
• $so(X)$: for null strings (eventually sent to zero in group case).

Dikranjan–Giordano Bruno–Virili proved a zero–infinity dichotomy: for any endomorphism, each type of string number is either $0$ or $\infty$; the existence of a single string guarantees infinitely many disjoint ones (Dikranjan et al., 2010). For endomorphisms of abelian groups, generalized shifts satisfy

$h_{alg}(\sigma_X) = s(X)\cdot\log|K|$

where $K$ is a finite abelian group and $s(X)$ the string number of the shift (Akhavin et al., 2010). This ties the algebraic entropy directly to combinatorial dynamics.

5. Entropy in Nonabelian and Lie Group Contexts

While the classical theory is most complete for abelian or torsion abelian cases, there are significant extensions to nonabelian groups and Lie groups. For torsion nilpotent groups, as discussed above, the Addition Theorem holds in full generality (Shlossberg, 14 Jan 2026). In the field of Lie groups, Caldas–Patrão established that for connected nilpotent or reductive Lie groups, the (topological) entropy of any surjective endomorphism equals the entropy of its restriction to the toral component of the center $T(G)$. For semisimple Lie groups, all such entropy vanishes (Caldas et al., 2011). On compact abelian or $p$-adic analytic groups, this reduces entropy calculations to the toral (abelian) case where they are governed by the spectrum of the induced linear transformation (1711.02562, Gunn et al., 2021).

In the context of simple abelian varieties over $\mathbb C$, entropy is dictated by the eigenvalues of the analytic representation on $H^{1,0}(X)$, with the entropy expressible as

$$h(\varphi) = \sum_{|\lambda_i|>1} 2\log|\lambda_i|,$$

and precise characterization of zero versus positive entropy in terms of the field structure of $\mathrm{End}^0(X)$ (Herrig, 2017).

6. Categorical, Adjoint, and Structural Aspects

Viewed from a categorical perspective, algebraic entropy is an entropy function on the abelian category of group flows. Satisfying invariance, vanishing on the zero object, (often) additive extension, and continuity under direct sums, such entropy functions define a corresponding Pinsker radical: the largest $\varphi$-invariant subobject/subgroup of zero entropy, forming a hereditary torsion theory (Dikranjan et al., 2010). This categorical framework unifies algebraic entropy, set-theoretic entropy, and their variants.

Adjoint algebraic entropy, defined via finite-index subgroups and cotrajectories, displays an even more rigid dichotomy: for any abelian group $G$ and endomorphism $\varphi$, the adjoint entropy $\mathrm{ent}^*(\varphi)$ is always $0$ or $\infty$ (Dikranjan et al., 2010). Via Pontryagin duality, adjoint entropy is identified with standard algebraic entropy of the adjoint map. For compact groups, this leads to sharp discontinuity criteria: no continuous endomorphism can have finite positive algebraic entropy.

7. Open Problems and Future Directions

Major ongoing directions include:

• Extending Addition Theorems to broader nonabelian classes (beyond torsion nilpotent and certain solvable cases) (Shlossberg, 14 Jan 2026).
• Understanding entropy and the structure of the Pinsker radical in locally compact or noncommutative settings (Dikranjan et al., 2010, Dikranjan et al., 2013).
• Investigating the interplay between entropy values and number-theoretic invariants (e.g., Mahler measure, Lehmer's problem), especially in connection with the algebraic Yuzvinski formula for entropy on vector spaces over $\mathbb Q$ or local fields (Dikranjan et al., 2010, Dikranjan et al., 2013).
• Combinatorial characterization of entropy for generalized shifts and connections to dynamical, set-theoretic, and growth invariants (Akhavin et al., 2010, Dikranjan et al., 2010).
• Exploring the dichotomies and rigidity in adjoint and categorical entropy, and seeking conceptual bridges to topological entropy in semiabelian categories and duality frameworks.

These strands integrate deep combinatorial, categorical, and number-theoretic aspects of group dynamics in the framework of entropy computations and structural decomposition.