Algebraic entropy for amenable semigroup actions

Anna Giordano Bruno; Antongiulio Fornasiero; Dikran Dikranjan

Algebraic entropy for amenable semigroup actions

T0 review · 0 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper defines algebraic entropy for actions of cancellative right amenable semigroups on abelian groups and proves that, for torsion groups, entropy is additive over invariant subgroups and equals the topological entropy of the dual…

desk verdict A robust generalization of algebraic entropy to amenable semigroup actions; the Addition and Bridge Theorems hold up under scrutiny and the paper deserves serious refereeing. read the letter →

arxiv 1908.01983 v1 pith:P5XSFEFP submitted 2019-08-06 math.GR math.DS

Dikran Dikranjan , Antongiulio Fornasiero , Anna Giordano Bruno This is my paper · ORCID

classification math.GRmath.DS MSC 20K3020M2037A3537B4043A07

keywords algebraicentropyamenablesemigroupactionsFølnernetsAdditionTheoremBridgePontryagindualitytopologicaltorsionabeliangroups

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper defines algebraic entropy—a measure of how fast finite sets grow under a group action—for left actions $S\curvearrowright A$ of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms, generalizing the classical case $S=\mathbb N$. Its central claim is an Addition Theorem: whenever $A$ is torsion and $B$ is an $\alpha$-invariant subgroup, $\operatorname{ent}(\alpha)=\operatorname{ent}(\alpha_B)+\operatorname{ent}(\alpha_{A/B})$. Its second central claim is a Bridge Theorem: for actions of cancellative left amenable monoids on totally disconnected compact abelian groups, the topological entropy of the action equals the algebraic entropy of the Pontryagin-dual action. These two theorems together yield an Addition Theorem for the topological entropy. The reader should care because the paper shows that entropy behaves like a length function on a natural algebraic category, and because the bridge makes algebraic and topological entropy computable from each other.

What carries the argument

The central object is the functional $H_S(f)=\lim_i f(F_i)/|F_i|$ on increasing, subadditive, left-subinvariant functions on the finite subsets of a cancellative right amenable monoid $S$, where $(F_i)$ is a right Følner net; its net-independence is imported from the semigroup mean theorem (Theorem 3.1). Applied to $f_X(F)=\log|\sum_{s\in F}\alpha(s)(X)|$, it produces $H_{\mathrm{alg}}(\alpha,X)$. The proof of the Addition Theorem is carried by the Filling Theorem (Theorem 6.6), which for any $\varepsilon$ finds finitely many translates $P_jF_j$ that almost cover a Følner set $D$ and are almost disjoint, so entropy sums split over the pieces; a Fubini-type theorem (Theorem 3.10) then controls the comparison of entropies over different right Følner nets. The Bridge Theorem is carried by Pontryagin duality: the annihilator correspondence $B\mapsto B^\perp$ bijects finite subgroups of the torsion discrete group with open subgroups of its compact dual, converting trajectory size $\log|T_F(B)|$ into the index $\log[\widehat A : C_F(\widehat\alpha,B^\perp)]$ that defines topological entropy.

What would settle it

Compute the defining trajectory limits for the $\mathbb N$-action on $A=\bigoplus_{i\in\mathbb Z}(\mathbb Z/2\mathbb Z)$ with shifts $e_i\mapsto e_{i+s}$, taking $B=\mathrm{span}\{e_i:i\ge 0\}$. The Addition Theorem predicts $\operatorname{ent}(\alpha)=\log 2$, $\operatorname{ent}(\alpha_B)=\log 2$, and $\operatorname{ent}(\alpha_{A/B})=0$; a direct count gives $|T_n(\{e_0\})|=2^n$ and $|T_n(\{e_{-1}+B\})|=2$. Any mismatch in the identity, or any dependence of the limits on the chosen right Følner net, would refute Theorem 1.1 or the imported mean theorem.

Watch

Extended reading notes

Core claim

The paper introduces two variants of algebraic entropy, $\operatorname{ent}$ and $h_{\mathrm{alg}}$, both defined by taking the supremum, over finite subgroups (for $\operatorname{ent}$) or finite subsets (for $h_{\mathrm{alg}}$), of the Følner average of $\log|T_F(\alpha,X)|$ along right Følner nets; the two coincide on torsion abelian groups. Its principal claim is that on torsion abelian groups this entropy is additive over invariant subgroups (Theorem 1.1), the proof of which uses a Filling Theorem to break an arbitrary Følner set into finitely many translates of fixed shapes, avoiding the classification of torsion abelian groups. Its second principal claim is the Bridge Theorem (Theorem 1.2): if $S$ is a cancellative left amenable monoid acting by continuous endomorphisms on a totally disconnected compact abelian group $K$, then the topological entropy of the action equals the algebraic entropy of the induced right action on the discrete Pontryagin dual $\widehat K$; since the duals of totally disconnected compact groups are exactly the torsion abelian groups, this connects the two entropies on the full class where both are defined.

Load-bearing premise

The load-bearing premise is the imported theorem that subadditive left-subinvariant functions on cancellative right amenable semigroups have a Følner-net-independent mean; if that mean depended on the net, neither entropy would be well-defined.

Editorial extensions

If this is right

For every left action of a cancellative right amenable monoid on a torsion abelian group, entropy is additive over invariant subgroups, making $\operatorname{ent}$ a length function on the category of torsion $\mathbb Z[S]$-modules that is also continuous under direct limits.
For cancellative left amenable monoids acting on totally disconnected compact abelian groups, topological entropy satisfies the same addition identity; this is a direct corollary of the Addition and Bridge Theorems.
Finite-index subgroups obey a Logarithmic Law: if $[G:H]=k$, then $h_{\mathrm{alg}}(\alpha\!\upharpoonright\!H)=k\,h_{\mathrm{alg}}(\alpha)$, and the same holds for $\operatorname{ent}$.
Many natural higher-rank actions have zero algebraic entropy: actions of $\mathbb N^d$ ($d>1$) on torsion-free finite-rank abelian groups, and the multiplicative action of upper-triangular subgroups of $\mathrm{GL}_n(K)$ on $K^n$ for infinite $K$.
Whenever an action factors through a quotient by an infinite normal subgroup lying in the kernel, its algebraic entropy is zero.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Filling-Theorem proof does not use the structure theory of torsion abelian groups, so the same template may prove the Addition Theorem for any additive invariant that is continuous under direct limits and behaves like a length function on module categories—beyond the logarithmic one used here.
The conjectures in Section 8 suggest the results should survive when $A$ is not torsion (Addition Theorem) and when $K$ is not totally disconnected (Bridge Theorem); a positive answer would make algebraic and topological entropy of amenable-group actions on locally compact abelian groups completely parallel.
The frequent vanishing for $\mathbb N^d$ and matrix group actions indicates that for genuinely multidimensional amenable semigroup actions, nonzero algebraic entropy is concentrated in shift-type actions and in phenomena invisible to the Følner average, a setting where alternative invariants such as receptive entropies would be tested.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The two things you should know: this is a genuinely substantial generalization of algebraic entropy from N-actions and amenable group actions to cancellative right amenable semigroup actions, and the core arguments are coherent. The Addition Theorem (ent for torsion abelian groups) and the Bridge Theorem (topological entropy equals algebraic entropy of the dual) are the real payoffs, and they appear to be proved correctly. The stress-test note is right: no load-bearing flaw surfaced on a careful pass, and the reliance on the imported Ornstein–Weiss/Filling machinery from [11] is legitimate external support, not a hidden assumption.

What the paper does well is more than just state two big theorems. The Fubini-type theorem for the integral of subadditive functions (Theorem 3.10) is a genuinely useful tool, and the proof of the Addition Theorem deliberately avoids the heavy structure theory of abelian groups used in earlier N-action proofs. The authors are also honest about what they cannot do: the Addition Theorem is proved for torsion groups only, and the Bridge Theorem for totally disconnected compact abelian groups; both limitations are stated explicitly, and the conjectural extensions are flagged as such. The citation pattern is fine—self-citations go to prior work that establishes the classical cases, and they are not used to force the new theorems.

Soft spots are local and repairable. The paper is dense, and the Følner machinery makes full verification painful; I could not check every epsilon. The minor issues the stress-test flagged—the constant choice in Proposition 2.17 and the unhandled Y = {0} case in Proposition 6.13—are exactly that: minor, and they do not threaten the main results. Proposition 6.13's proof would benefit from a sentence treating the trivial quotient, but that is a detail. A referee should also push on whether the integral framework in Section 3 can be simplified, since the present exposition is heavier than the underlying ideas require.

This paper is for researchers in algebraic entropy, amenable group actions, and topological entropy on abelian groups. It will likely become standard machinery in that niche. I would cite it and would bring it to a reading group. The right call is to send it to peer review: it deserves a serious referee, not a desk reject, even though the referee will need to do some work to verify the technical core.

Referee Report

0 major / 7 minor

Summary. The paper extends algebraic entropy to left actions of cancellative right amenable monoids on discrete abelian groups by endomorphisms, defining two quantities, halg and ent, which agree on torsion groups. The main results are Theorem 1.1 (Addition Theorem): for a left action of a cancellative right amenable monoid on a torsion abelian group and an invariant subgroup B, ent(alpha)=ent(alpha_B)+ent(alpha_{A/B}); and Theorem 1.2 (Bridge Theorem): for a left action of a cancellative left amenable monoid on a totally disconnected compact abelian group, the topological entropy equals the algebraic entropy of the dual right action. The paper also develops a Følner-based integration theory for subadditive functions on amenable semigroups, proves restriction and quotient entropy formulas for amenable group actions, and derives an Addition Theorem for topological entropy of actions on totally disconnected compact abelian groups via Pontryagin duality.

Significance. If correct, the results unify and generalize the classical Weiss/Peters algebraic entropy and the corresponding Bridge Theorem to a substantial class of amenable semigroup actions, going beyond the group actions treated by Virili. The proofs are explicit and are built on the published Ornstein–Weiss-type theorem of Ceccherini-Silberstein–Coornaert–Krieger and the Filling Theorem; the main arguments do not circularly reduce to the quantities being defined, and no fitted parameters are used. The paper is a serious contribution to entropy theory on groups and monoids and should be of interest to readers in algebraic entropy, topological entropy, and amenability. The strengths include the careful statement of the external tools on which the construction depends and the explicit epsilon-tracking in the proof of the Addition Theorem, modulo the presentation issues noted below.

minor comments (7)

[2.3, Proposition 2.17] The proof invokes Lemma 2.7(a) to find F with |FE \ F|/|F| ≤ 1/n, but the defining property (2.6) requires |F(E,n)s Δ F(E,n)|/|F(E,n)| ≤ 1/n for each s ∈ E. The proof should instead use |FE Δ F|/|F| ≤ 1/n (or adjust the constant by a factor of 2) so that the stated definition of a canonically indexed right Følner net is actually satisfied.
[6.3, Eqs. (6.7) and (6.13)] The displayed inequalities in the proof of Proposition 6.13 use what appears to be multiplicative epsilons, e.g., '≤5ε ent(αB) + Halg(...)' and '≤3ε Halg(...)'. The surrounding derivation supports only additive epsilon terms of the form '≤5ε + ent(αB) + Halg(...)' and '≤Halg(...) + 3ε'. These displays should be corrected to match the argument, since the literal multiplicative reading is inconsistent with the rest of the proof.
[6.3, Proposition 6.13] The definition of \barε = ε/(2ℓ(Y)) is undefined when Y = {0}, since ℓ(Y) = 0. The case Y = {0} should be treated separately (its entropy contribution is zero) so that the proof covers all finite subgroups Y of A.
[6.3, Proposition 6.13] The hypothesis is stated as 'S a right amenable monoid', but the proof applies Corollary 6.10 and Theorem 6.6, both of which require cancellativity of S. The statement should read 'cancellative right amenable monoid' in order for the quoted external theorems to apply.
[1, Theorem 1.2] The statement says that γ induces a dual right action on 'the Pontryagin dual of A', but the group A has not yet been introduced at that point; the intended group is the Pontryagin dual of K, namely A = \hat{K}. This should be corrected to avoid confusion.
[5.1, proof of Lemma 5.2] The expression 'Halg(α, ↾G,TEm(α↾H,X))' contains a misplaced comma and should be written as Halg(α↾G, T_{E_m}(α↾H,X)) for clarity.
[Throughout] The manuscript contains numerous typographical and OCR artifacts, such as 'ε2N' in Theorem 6.6, incomplete or garbled footnote material in §2.3, and inconsistent notation like 'δn,n' in Example 2.28. These should be cleaned up before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Addition and Bridge Theorems are proved from external Følner machinery and Pontryagin duality, not from their own conclusions.

full rationale

The definitions of ent, halg, and htop all depend on the net-independence of Følner limits, which is imported from the external Ornstein–Weiss-type theorem of Ceccherini-Silberstein, Coornaert, and Krieger (Theorem 3.1 and Corollary 3.2, from [11]). This is a legitimate external foundation and not a self-citation. The Addition Theorem (Theorem 1.1) is proved in Propositions 6.12 and 6.13 using the imported Filling Theorem 6.6 to construct ε-tilings; the estimates compare trajectory sizes directly and the target equality is never assumed as a hypothesis. The Bridge Theorem (Theorem 1.2, proved in Theorem 7.7) rests on the pointwise identity |T_F(α,B)| = [Â : C_F(α̂,B⊥)] of Lemma 7.6, obtained from annihilators and Pontryagin duality, and then on taking the same Følner limit on both sides; this is a genuine transfer theorem rather than a definitional reduction. Self-citations to earlier papers such as [28] and [30] are invoked only to recall established facts about the classical N-action case in examples and corollaries, not to force the new central theorems. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' own work to close an argument, and no ansatz is smuggled in by citation. The derivation chain is therefore self-contained relative to its stated external foundations.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central results rest on external Følner machinery, classical duality, and the explicit structural assumptions listed here. No free parameters are fitted and no new postulated entities are introduced.

assumptions (8)

standard math Ornstein-Weiss Lemma for cancellative right amenable semigroups: every subadditive, left subinvariant, uniformly bounded function on finite subsets has a Følner-net-independent mean.
Imported from [11, Theorem 1.1] as Theorem 3.1 and Corollary 3.2; it is the foundation of the entropy integral used throughout Section 4.
standard math Existence and characterization of Følner nets for cancellative right amenable monoids.
Derived from [59, Corollary 4.3] in Theorem 2.4; used to construct canonically indexed Følner nets and to transfer Følner properties to products and sections.
standard math Filling Theorem for ε-tilings of Følner sets in cancellative semigroups.
Imported from [11, Theorem 3.8] as Theorem 6.6; it is load-bearing in the proof of the harder inequality of the Addition Theorem in Proposition 6.13.
standard math Pontryagin-van Kampen duality, annihilator properties, and van Dantzig's theorem that open subgroups form a local base for totally disconnected compact groups.
These classical facts underlie the Bridge Theorem by identifying finite subgroups of the dual with open subgroups, as in Section 7.
standard math Right amenable semigroups satisfy the left Ore condition and admit common right multiples.
Used in Lemma 2.24 and Corollary 4.30; cited from [63, Proposition 1.23].
domain assumption Domain assumption: S is a cancellative right amenable monoid for algebraic entropy and a cancellative left amenable monoid for topological entropy.
All definitions and main theorems are stated under this amenability and cancellativity hypothesis; moving outside these classes would require different entropy notions.
domain assumption Good section hypothesis in Theorem 3.10: a quotient map π:S→C admits a section σ whose values are good elements, so Nσ(c)=σ(c)N=π^{-1}(c).
This hypothesis is not automatic, as Examples 2.21(e)-(g) show; it is needed for the Fubini-type theorem and for the quotient entropy computations in Section 5.
domain assumption A is a torsion abelian group in the Addition Theorem and the Bridge Theorem.
Both theorems are stated for torsion abelian groups; the general non-torsion case is left as Conjectures 8.5 and 8.6.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Algebraic entropy for amenable semigroup actions." pith.science (2026). https://pith.science/paper/P5XSFEFP

@misc{pith2026190801983,
  author       = {Pith},
  title        = {Pith review of: Algebraic entropy for amenable semigroup actions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P5XSFEFP}},
  note         = {Machine review of arXiv:1908.01983}
}

abstract

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case $S=\mathbb N$. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of the Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages

[11]

Ceccherini-Silberstein, M

T. Ceccherini-Silberstein, M. Coornaert, F. Krieger, An analogue of Fekete’s lemma for subadditive functions on cancellative amenable semigroups , J. Anal. Math. 124 (2014) 59–81

work page 2014
[1]

Adler, A

R. Adler, A. Konheim, M. McAndrew, Topological entropy, Trans. AMS 114 (1965) 309–319

work page 1965
[2]

Akhavin, F

M. Akhavin, F. Ayatollah Zadeh Shirazi, D. Dikranjan, A. Giordano Bruno, A. Hosseini, Algebraic entropy of shift endomorphisms on abelian groups , Quaest. Math. 32 (2009) 529–550

work page 2009
[3]

Aoki, Topological entropy and measure-theoretic entropy for aut omorphisms on compact groups , Math

N. Aoki, Topological entropy and measure-theoretic entropy for aut omorphisms on compact groups , Math. Systems Theory 5 (1971) 4–7

work page 1971
[4]

Ayatollah Zadeh Shirazi, D

F. Ayatollah Zadeh Shirazi, D. Dikranjan, Set-theoretical entropy: A tool to compute topological ent ropy, Proceed- ings Islamabad ICTA 2011, Cambridge Scientiﬁc Publishers 2012, 11– 32

work page 2011
[5]

B ´ ıs, D

A. B ´ ıs, D. Dikranjan, A. Giordano Bruno, L. Stoyanov, Topological entropy, upper capacity and fractal dimension s of ﬁnitely generated semigroup actions , submitted

work page
[6]

B ´ ıs, D

A. B ´ ıs, D. Dikranjan, A. Giordano Bruno, L. Stoyanov, Metric entropy for group and semigroup actions , preprint

work page
[7]

Bowen, Entropy for group endomorphisms and homogeneous spaces , Trans

R. Bowen, Entropy for group endomorphisms and homogeneous spaces , Trans. Amer. Math. Soc. 153 (1971) 401– 414

work page 1971

Show all 78 references

[8]

Bowen, Measure conjugacy invariants for actions of countable soﬁc groups, J

L. Bowen, Measure conjugacy invariants for actions of countable soﬁc groups, J. Amer. Math. Soc. 23 (2010) 217–245

2010
[9]

Bowen, Soﬁc entropy and amenable groups , Ergodic Theory Dynam

L. Bowen, Soﬁc entropy and amenable groups , Ergodic Theory Dynam. Systems 32 (2012) 427–466

2012
[10]

Ceccherini-Silberstein, M

T. Ceccherini-Silberstein, M. Coornaert, Cellular automata and groups , Springer Monographs in Mathematics, Springer-Verlag Berlin, 2010

2010
[12]

Cliﬀord, G

A. Cliﬀord, G. Preston, The algebraic theory of semigroups , Vol. I. Mathematical Surveys, No. 7 American Math- ematical Society Providence R.I., 1961

1961
[13]

Cliﬀord, G

A. Cliﬀord, G. Preston, The algebraic theory of semigroups , Vol. II. Mathematical Surveys, No. 7 American Mathematical Society Providence R.I., 1967

1967
[14]

J. P. Conze, Entropie d’un groupe ab´ elien de transformations, Z. Wahrscheinlichkeitstheorie und verwandte Gebiete 25 (1972) 11–30

1972
[15]

Chung, A

N. Chung, A. Thom, Some remarks on the entropy for algebraic actions of amenabl e groups , Trans. Amer. Math. Soc. 367 (2015) 8579–8595

2015
[16]

M. M. Day, Means for the bounded functions and ergodicity of the bounde d representations of semigroups , Trans. Amer. Math. Soc. 69 (1950) 276–291

1950
[17]

M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957) 509–544

1957
[18]

M. M. Day, Normed linear spaces , Academic Press, Inc. Publishers New York, Springer-Verlag Berlin -G¨ ottingen- Heidelberg, 1962

1962
[19]

M. M. Day, Semigroups and amenability , in Semigroups, Proc. Sympos., Wayne State Univ., Detroit, Mich., 19 68, Academic Press New York, 1969, 5–53

1969
[20]

Deninger, Fuglede-Kadison determinants and entropy for actions of di screte amenable groups , J

C. Deninger, Fuglede-Kadison determinants and entropy for actions of di screte amenable groups , J. Amer. Math. Soc. 19 (2006) 737–758

2006
[21]

Dikranjan, A

D. Dikranjan, A. Fornasiero, A. Giordano Bruno, Entropy of generalized shifts and related topics , work in progress
[22]

Dikranjan, A

D. Dikranjan, A. Fornasiero, A. Giordano Bruno, A short proof of the addition formula for the algebraic entro py, preprint
[23]

Dikranjan, A

D. Dikranjan, A. Giordano Bruno, The Pinsker subgroup of an algebraic ﬂow , J. Pure Appl. Algebra (2012) 364–376

2012
[24]

Dikranjan, A

D. Dikranjan, A. Giordano Bruno, Topological and algebraic entropy on groups , Proceedings Islamabad ICTA 2011, Cambridge Scientiﬁc Publishers 2012, 133–214

2011
[25]

Dikranjan, A

D. Dikranjan, A. Giordano Bruno, The connection between topological and algebraic entropy , Topology Appl. 159 (2012) 2980–2989

2012
[26]

Dikranjan, A

D. Dikranjan, A. Giordano Bruno, Discrete dynamical systems in group theory , Note Mat. 33 (2013) 1–48

2013
[27]

Dikranjan, A

D. Dikranjan, A. Giordano Bruno, The Bridge Theorem for totally disconnected LCA groups , Topology Appl. 169 (2014) 21–32

2014
[28]

Dikranjan, A

D. Dikranjan, A. Giordano Bruno, Entropy on abelian groups , Adv. Math. 298 (2016) 612–653

2016
[29]

Dikranjan, A

D. Dikranjan, A. Giordano Bruno, Entropy on normed semigroups , to appear in Dissertationes Mathematicae
[30]

Dikranjan, B

D. Dikranjan, B. Goldsmith, L. Salce, P. Zanardo, Algebraic entropy for abelian groups , Trans. Amer. Math. Soc. 361 (2009) 3401–3434

2009
[31]

Dikranjan, M

D. Dikranjan, M. Sanchis, Bowen’s entropy for endomorphisms of totally bounded abeli an Groups , Descriptive Topology and Functional Analysis, Springer Proceedings in Mathema tics & Statistics, Volume 80 (2014) 143–162

2014
[32]

Dikranjan, M

D. Dikranjan, M. Sanchis, Dimension and entropy in compact topological groups , Journal of Mathematical Analysis and its Applications 476, Issue 2, 15 (2019) 337–366

2019
[33]

Dikranjan, M

D. Dikranjan, M. Sanchis, S. Virili, New and old facts about entropy in uniform spaces and topolog ical groups , Topology Appl. 159 (2012) 1916–1942

2012
[34]

Dinaburg, On the relations among various entropy characteristics of d ynamical systems, Izv

E. Dinaburg, On the relations among various entropy characteristics of d ynamical systems, Izv. Akad. Nauk SSSR 35 (1971) 324–366 (Math. USSR Izvestija 5 (1971) 337–378)

1971
[35]

Donnelly, Subsemigroups of Cancellative Amenable Semigroups , Int

J. Donnelly, Subsemigroups of Cancellative Amenable Semigroups , Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 23, 1131–1137. 45

2012
[36]

Følner, On groups with full Banach mean value , Math

E. Følner, On groups with full Banach mean value , Math. Scand. 3 (1995) 243–254

1995
[37]

A. H. Frey Jr, Studies on amenable semigroups , ProQuest LLC, Ann Arbor, MI, 1960; Ph.D. Thesis, University of Washington

1960
[38]

Giordano Bruno, Algebraic entropy of shift endomorphisms on products , Communications in Algebra 38 (2010) 4155–4174

A. Giordano Bruno, Algebraic entropy of shift endomorphisms on products , Communications in Algebra 38 (2010) 4155–4174

2010
[39]

Giordano Bruno, A Bridge Theorem for the entropy of semigroup actions , submitted

A. Giordano Bruno, A Bridge Theorem for the entropy of semigroup actions , submitted
[40]

Giordano Bruno, P

A. Giordano Bruno, P. Spiga, Some properties of the growth and of the algebraic entropy of group endomorphisms, J. Group Theory 20 (4) (2017) 763–774

2017
[41]

Giordano Bruno, P

A. Giordano Bruno, P. Spiga, Milnor-Wolf ’s Theorem for group endomorphisms , submitted
[42]

Giordano Bruno, S

A. Giordano Bruno, S. Virili, Algebraic Yuzvinski Formula, J. Algebra 423 (2015) 114–147

2015
[43]

Giordano Bruno, S

A. Giordano Bruno, S. Virili, On the Algebraic Yuzvinski Formula , Topol. Algebra and its Appl. 3 (2015) 86–103

2015
[44]

Goldsmith, L

B. Goldsmith, L. Salce, Algebraic entropies for Abelian groups with applications t o the structure of their endomor- phism rings: a survey , Groups, modules, and model theory–surveys and recent develo pments, 135–174, Springer, Cham, 2017

2017
[45]

Goldsmith, L

B. Goldsmith, L. Salce, Corner’s realization theorems from the viewpoint of algebr aic entropy, Rings, polynomials, and modules, 237–255, Springer, Cham, 2017

2017
[46]

K. H. Hofmann, L. Stoyanov, Topological entropy of group and semigroup actions , Adv. Math. 115 (1995) 54-98

1995
[47]

Katznelson, B

Y. Katznelson, B. Weiss, Commuting measure-preserving transformations , Israel J. Math. 12 (1972) 161–173

1972
[48]

D. Kerr, H. Li, Entropy and the variational principle for actions of soﬁc gr oups, Invent. Math. 186 (2011) 501–558

2011
[49]

A. A. Kirillov, Dynamical systems, factors and group representations , Russ. Math. Surv. 22 (1967) 67–80

1967
[50]

R. D. Gray, M. Kambites, Amenability and geometry of semigroups , Trans. Amer. Math. Soc. 369 (2017) 8087– 8103

2017
[51]

Hindman, D

N. Hindman, D. Strauss, Density and invariant means in left amenable semigroups , Topology Appl. 156 (2009) 2614–2628

2009
[52]

Hochster, Subsemigroups of amenable groups , Proc

M. Hochster, Subsemigroups of amenable groups , Proc. Amer. Math. Soc. 21 (1969) 363–364

1969
[53]

Kieﬀer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space , Ann

J.C. Kieﬀer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space , Ann. Probability 3 (1975) 1031–1037

1975
[54]

Klawe, Semidirect product of semigroups in relation to amenabilit y, cancellation properties, and strong Folner conditions, Paciﬁc J

M. Klawe, Semidirect product of semigroups in relation to amenabilit y, cancellation properties, and strong Folner conditions, Paciﬁc J. Math. 73 (1977) 91–106

1977
[55]

Klawe, Dimensions of the sets of invariant means of semigroups , Illinois J

M. Klawe, Dimensions of the sets of invariant means of semigroups , Illinois J. Math. 24 (1980) 233–243

1980
[56]

Li, Compact group automorphisms, addition formulas and Fugled e-Kadison determinants , Ann

H. Li, Compact group automorphisms, addition formulas and Fugled e-Kadison determinants , Ann. of Math. (2) 176 (2012) 303–347

2012
[57]

H. Li, B. Liang, Soﬁc mean length , to appear in Adv. Math
[58]

D. Lind, K. Schmidt, T. Ward, Mahler measure and entropy for commuting automorphisms of c ompact groups , Invent. Math. 101 (1990) 593–629

1990
[59]

Namioka, Følner’s conditions for amenable semigroups , Math

I. Namioka, Følner’s conditions for amenable semigroups , Math. Scand. 15 (1964) 18–28

1964
[60]

Northcott, M

D.G. Northcott, M. Reufel, A generalization of the concept of length , Q. J. Math. Oxford (2) 16 (1965) 297–321

1965
[61]

Ollagnier, Ergodic theory and statistical mechanics , Lecture Notes in Math

J.M. Ollagnier, Ergodic theory and statistical mechanics , Lecture Notes in Math. 1115, Springer Verlag, Berlin- Heidelberg-New York, 1985

1985
[62]

Ornstein, B

D. Ornstein, B. Weiss, Entropy and isomorphism theorems for actions of amenable gr oups, J. Analyse Math. 48 (1987) 1–141

1987
[63]

A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29. American Mathemat ical Society, Providence, RI, 1988

1988
[64]

Peters, Entropy on discrete abelian groups , Adv

J. Peters, Entropy on discrete abelian groups , Adv. Math. 33 (1979) 1–13

1979
[65]

Peters, Entropy of automorphisms on LCA groups , Paciﬁc J

J. Peters, Entropy of automorphisms on LCA groups , Paciﬁc J. Math. 96 (1981) 475–488

1981
[66]

Salce, P

L. Salce, P. Vamos, S. Virili, Length functions, multiplicities and algebraic entropy , Forum Math. 25 (2013) 255– 282

2013
[67]

Salce, P

L. Salce, P. Zanardo, A general notion of algebraic entropy and the rank-entropy , Forum Math. 21 (2009) 579–599

2009
[68]

Schmidt, Dynamical systems of algebraic origin , Progr

K. Schmidt, Dynamical systems of algebraic origin , Progr. Math. 128, Birkh¨ a user Zentralblatt Verlag, Basel, 1995

1995
[69]

A. M. Stepin, A. T. Tagi-Zade, Variational characterization of topological pressure for amenable groups of trans- formations, Dokl Akad. Nauk SSSR 254 (1980) 545–549

1980
[70]

L. N. Stoyanov, Uniqueness of topological entropy for endomorphisms on com pact groups, Boll. Un. Mat. Ital. B (7) 1 (1987) 829–847

1987
[71]

V´ amos, Additive functions and duality over Noetherian rings , Q

P. V´ amos, Additive functions and duality over Noetherian rings , Q. J. Math. (Oxford) (2) 19 (1968) 43–55

1968
[72]

Virili, Entropy for endomorphisms of LCA groups , Toplogy Appl

S. Virili, Entropy for endomorphisms of LCA groups , Toplogy Appl. 159 (2012) 2546–2556

2012
[73]

Virili, Algebraic entropy of amenable group actions , Math

S. Virili, Algebraic entropy of amenable group actions , Math. Z. 291 (2019) 1389–1417

2019
[74]

Virili, Algebraic and topological entropy of group actions , preprint

S. Virili, Algebraic and topological entropy of group actions , preprint
[75]

Weiss, Monotileable amenable groups , Topology, ergodic theory, real algebraic geometry, 257–262, A mer

B. Weiss, Monotileable amenable groups , Topology, ergodic theory, real algebraic geometry, 257–262, A mer. Math. Soc. Transl. Ser. 2, 202, Adv. Math. Sci., 50, Amer. Math. Soc., Pr ovidence, RI, 2001

2001
[76]

Weiss, Entropy and actions of soﬁc groups , Discrete Contin

B. Weiss, Entropy and actions of soﬁc groups , Discrete Contin. Dyn. Syst. Ser. B 20 (2015) 3375–3383

2015
[77]

M. D. Weiss, Algebraic and other entropies of group endomorphisms , Math. Systems Theory 8 (3) (1974/75) 243–248

1974
[78]

Yuzvinski, Metric properties of endomorphisms of compact groups , Izv

S. Yuzvinski, Metric properties of endomorphisms of compact groups , Izv. Acad. Nauk SSSR, Ser. Mat. 29 (1965) 1295-1328 (in Russian); English Translation: Amer. Math. Soc. Tra nsl. 66 (1968) 63–98. 46

1965

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

Ceccherini-Silberstein, M

T. Ceccherini-Silberstein, M. Coornaert, F. Krieger, An analogue of Fekete’s lemma for subadditive functions on cancellative amenable semigroups , J. Anal. Math. 124 (2014) 59–81

work page 2014

[2] [1]

Adler, A

R. Adler, A. Konheim, M. McAndrew, Topological entropy, Trans. AMS 114 (1965) 309–319

work page 1965

[3] [2]

Akhavin, F

M. Akhavin, F. Ayatollah Zadeh Shirazi, D. Dikranjan, A. Giordano Bruno, A. Hosseini, Algebraic entropy of shift endomorphisms on abelian groups , Quaest. Math. 32 (2009) 529–550

work page 2009

[4] [3]

Aoki, Topological entropy and measure-theoretic entropy for aut omorphisms on compact groups , Math

N. Aoki, Topological entropy and measure-theoretic entropy for aut omorphisms on compact groups , Math. Systems Theory 5 (1971) 4–7

work page 1971

[5] [4]

Ayatollah Zadeh Shirazi, D

F. Ayatollah Zadeh Shirazi, D. Dikranjan, Set-theoretical entropy: A tool to compute topological ent ropy, Proceed- ings Islamabad ICTA 2011, Cambridge Scientiﬁc Publishers 2012, 11– 32

work page 2011

[6] [5]

B ´ ıs, D

A. B ´ ıs, D. Dikranjan, A. Giordano Bruno, L. Stoyanov, Topological entropy, upper capacity and fractal dimension s of ﬁnitely generated semigroup actions , submitted

work page

[7] [6]

B ´ ıs, D

A. B ´ ıs, D. Dikranjan, A. Giordano Bruno, L. Stoyanov, Metric entropy for group and semigroup actions , preprint

work page

[8] [7]

Bowen, Entropy for group endomorphisms and homogeneous spaces , Trans

R. Bowen, Entropy for group endomorphisms and homogeneous spaces , Trans. Amer. Math. Soc. 153 (1971) 401– 414

work page 1971