Recurrence and Entropy for Discrete-Time Deterministic Dynamical Systems

Round 1

Reviewer 1 Report (Previous Reviewer 2)

Comments and Suggestions for Authors

Accept in present form.

Author Response

Reviewer 1 required no further edition. 

Reviewer 2 Report (Previous Reviewer 3)

Comments and Suggestions for Authors

Recommend for publication

Author Response

Reviewer 2 required no further revisions.

Reviewer 3 Report (Previous Reviewer 4)

Comments and Suggestions for Authors

This paper attempts to establish the relationship between structural constraints and information entropy by introducing the symmetry of countable ordered groups and assuming strong resilience ( no positive measure weak wandering set ) in the deterministic dynamic system on finite state space. The research topic is at the intersection of dynamic system, ergodic theory and information theory, which has potential theoretical significance. However, the current version needs to be modified to address the following issues. 1. The paper attempts to integrate multiple concepts under different mathematical frameworks, but fails to clearly define them, resulting in contradictions and logical breaks. Confusion of state spaces ( finite vs.measure spaces ) : The abstract and introduction ( page 1 ) claim to study ' finite state spaces ', but the materials and methods ( section 2 ) immediately turn to a general ' σ-finite measure space ( Ω, B, m ) '. More confusingly, in Definition 2.8 and Theorem 3.22, an atomic and finite ' effective state space ' Ωeff is derived from the non-atomic measure space. The author tries to explain this shift in Remark 3.27, but the explanation is not sufficient. The properties of strong resilience in non-atomic measure spaces ( such as Proposition 3.11 ) and the simple periodicity on finite sets ( such as Example 3.1 ) are completely different mathematical phenomena. A group action defined on a general measure space is directly ' restricted ' or ' projected ' onto a finite set Ωeff defined by a single trajectory, and the invariant measure on it is discussed. This step has a huge jump and lacks strict mathematical proof. How to naturally and uniquely induce a group action on a finite set from a group action on a σ-finite measure space ? This is not clearly stated in the text. 2. Fuzziness of the definition of ' deterministic system ' ( Definition 2.1 ) : Definition It is claimed that the system is defined by the trajectory map X : G + → Ω, and is generated by a transition map T : Ω → Ω. However, the subsequent discussion on the role of group G ( Definition 2.3 ) requires the exchange of the role of T and G. Then, is T only the action of a specific generator in G + ( for example, ( 1,0 ) of Z2 ) ? If so, then the trajectory is just along a specific subsemigroup of G. What does this have to do with the symmetry of the whole group G ? In Example 3.24 ( ii ), the author himself also pointed out that only the orbit of T can only cover a part of the state space, and ' full G-action ' is needed to achieve transitivity. So, is the ' dynamics ' of the system an iteration of T, or the whole G-orbit ? This ambiguity directly affects the definition of Ωeff ( is it TT-or G-orbital ? ) and strong resilience application objects. 3.Definition and interpretation of ' entropy ' ( Definition 2.8 & Remark 2.9 ) : The author calls the defined quantity H ( X ) as ' deterministic Shannon entropy '. However, Remark 2.9 ( ii ) clearly states that p ( x ) is the ' time average access frequency '. In the case of transmission, this is indeed a uniform distribution, and the entropy is log | Ωeff |. But what does this have to do with Kolmogorov-Sinai entropy ( which the authors acknowledge as 0 in Remark 3.2 ( ii ) ) ? A deterministic, periodic finite system ( with KS entropy 0 ) is described as having ' maximum uncertainty ' or or ' configuration complexity ', rather than the ' unpredictability ' of its dynamics. A simple n-periodic cycle x ∂ x + 1 ( modn ) is described as having the maximum entropy logn, which is mathematically true, but in the context of dynamical systems, it is extremely inappropriate to equate it with the unpredictability of stochastic processes. The narrative of the thesis over-exaggerates the novelty and profundity of this result. 4. There is a serious flaw in the proof of the core theorem ( Theorem 3.22 ), Theorem 3.22 is the main result of the paper. But its proof ( page 16 ) has fatal flaws. Unique ergodicity misuse and cyclic argument : Proof claims depend on 'unique ergodicity 'and Folner sequences. However, the author assumes that the system ( G, Ω, m ) is ergodic and uniquely ergodic ( page 15 ). But what is the definition of ' unique ergodicity ' when the state space Ω is a general measure space ( may not be compact ) and there is no topological structure ? Usually, the unique ergodicity is for compact topological dynamical systems which preserve continuous transformations. More importantly, the conclusion that the author tries to prove is that ' the frequency converges to the G-invariant probability measure μ ', and the proof directly uses the unique ergodicity, which is tantamount to circular argument. The goal of the theorem should be to derive the convergence of the frequency and the uniformity of the distribution from strong resilience and symmetry, rather than as a prerequisite assumption. The authors try to avoid the pointwise ergodic theorem by weak convergence and compactness argument ( Remark 3.23 ( iii ) ), but it still needs to prove that all limit points are G-invariant and all Folner sequences converge to the same limit. This is a very strong conclusion for a non-compact pure measure space system lacking topological structure, which can not be easily obtained only by strong resilience. 5. The definition of group action on Ωeff is not clear : In the theorem statement, it is assumed that Ωeff is ' a finite effective state space invariant under G action '. However, how is the role of G on Ωeff defined ? If Ωeff is generated by only one T-trajectory ( a subsemigroup of G ), then the action of the whole group G on Ωeff may not be well defined at all. The authors need to prove that the role of G on Ω can be limited to Ωeff and remain transitive. For non-transitive cases ( e.g., Corollary 3.25 ), Ωeff ( x0 ) is more likely to be just a T-orbit. What is the G-interaction structure on it ? These key issues have not been answered. 6.Disconnection between Abstract and text : Abstract claims that the analysis is extended to 'the role of any countable linear ordered group ', but the main result of the text ( Theorem 3.22 ) is only for 'amenable groups ', which is an important limitation and is not mentioned in the abstract. The logical break between Section 2 and Section 3 : Section 2 introduces the strong resilience on general measure space, Prasad 's work and so on, but the core result of Section 3 finally settles on a finite, atomic set Ωeff. The key mathematical steps of how to transition from the former to the latter are missing. The supplementary explanation of Remark 3.27 is an ex post facto remedy, which is not enough to form a strict logical chain.

Author Response

REVIEWER 3

1. The paper attempts to integrate multiple concepts under different mathematical frameworks, but fails to clearly define them, resulting in contradictions and logical breaks. Confusion of state spaces (finite vs. measure spaces) : The abstract and introduction (page 1) claim to study 'finite state spaces', but the materials and methods (section 2) immediately turn to a general 'σ-finite measure space (Ω, B, m)'. More confusingly, in Definition 2.8 and Theorem 3.22, an atomic and finite 'effective state space' Ωeff is derived from the non-atomic measure space. The author tries to explain this shift in Remark 3.27, but the explanation is not sufficient. The properties of strong resilience in non-atomic measure spaces (such as Proposition 3.11) and the simple periodicity on finite sets (such as Example 3.1) are completely different mathematical phenomena. A group action defined on a general measure space is directly ' restricted ' or ' projected ' onto a finite set Ωeff defined by a single trajectory, and the invariant measure on it is discussed. This step has a huge jump and lacks strict mathematical proof. How to naturally and uniquely induce a group action on a finite set from a group action on a σ-finite measure space? This is not clearly stated in the text.

Response to Comment 1 (finite vs. σ-finite confusion; transition to Ωeff​):

• The abstract is rewritten to distinguish between the two levels of the architecture.
• A new "two-level architecture" is inserted immediately after Remark 2.2. It explicitly names the two levels: Level 1 (ambient σ-finite space where recurrence lives) and Level 2 (finite atomic space Ωeff​ where entropy is computed) and explains the "support-switch" from m to μ.
• The apparent contradiction between the non-atomic ambient space (Ω,B,m) and the finite atomic effective state space Ωeff​ is now addressed explicitly and systematically throughout the revised manuscript, rather than only in the concluding Remark 3.27 (Now Remark 3.29(iii)).
• The mathematical bridge between the two levels, is now made explicit in the road-map Remark 2.9(ii): At Level 1, the ambient space (Ω,m) is a σ-finite measure space, possibly non-atomic and infinite, and it is here that strong recurrence, wandering sets, and conservativity are formulated; these are genuinely measure-theoretic phenomena and the results of Sections 3.2–3.4 live entirely at this level, requiring no finiteness of Ωeff. At Level 2, when the G+-orbit of a given initial point x_0​ is finite (which is a property of the dynamics, not of the ambient space; it occurs, for instance, when the G+-action on Ω is periodic), we perform a deliberate switch: since m(Ωeff)=0, whenever m is non-atomic (individual trajectory points have zero ambient measure), the ambient measure m is abandoned on Ωeff​ and replaced by the canonical atomic probability measure μ defined by empirical visit frequencies along the trajectory, whose existence and uniqueness is the content of Theorem 3.22 (not an assumption). The Shannon entropy H(X) is then a property of the finite atomic space (Ωeff​,μ) alone, entirely independent of m.
• The properties of strong recurrence in non-atomic measure spaces such as Proposition 3.13 (formerly Proposition 3.11) and the simple periodicity on finite sets such as Example 3.1 are now clarified completely as different mathematical phenomena.
• The referee correctly asked about uniqueness of the induced action on Ωeff from the group action on σ-finite measure space Ω: This is clearly explained in the paragraph before Remark 2.3. The fact that the restricted action is well-defined is shown in Remark 2.3(iii).

2. Fuzziness of the definition of 'deterministic system' (Definition 2.1): It is claimed that the system is defined by the trajectory map X: G + → Ω, and is generated by a transition map T: Ω → Ω. However, the subsequent discussion on the role of group G requires the exchange of the role of T and G. Then, is T only the action of a specific generator in G+ (for example, (1,0) of Z^2)? If so, then the trajectory is just along a specific subsemigroup of G. What does this have to do with the symmetry of the whole group G? In Example 3.24 (ii), the author himself also pointed out that the orbit of T can only cover a part of the state space, and 'full G-action' is needed to achieve transitivity. So, is the 'dynamics' of the system an iteration of T, or the whole G-orbit? This ambiguity directly affects the definition of Ωeff (is it T-or G-orbital?) and strong resilience application objects.

Response to Comment 2. (ambiguity of T vs. G):

• Definition 2.1 is completely rewritten that explicitly states that the G-action is parametrized into a “time-evolution” via a semigroups (T_g)_{g\in G+} of maps.
• The trajectory map X: G+ → Ω, and transition map T: Ω → Ω are no longer needed and are dropped from the definition and all subsequent applications. All references to a single transition map T in the examples are dropped. All cases where T was only the action of a specific generator in G+ (for example, (1,0) of Z^2) are now dropped.
• Example 3.24 (ii) is corrected (now Example 3.26 (ii)). The comment that “the orbit of T can only cover a part of the state space, and 'full G-action' is needed to achieve transitivity” was wrong and is completely dropped.
• The whole system is called X now (see Definition 2.1) which is also used in the entropy notation H(X). All the references to a system (X_t)_{t\in G+}, which was confusing, are now dropped from the introduction, examples, and all the results.
• The commutativity condition T(g⋅x)=g⋅T(x)T(g\cdot x) = g\cdot T(x) T(g⋅x)=g⋅T(x) is no longer needed and is dropped. The fact that G acts well-defined on Ωeff in Remark 2.3(iii). It is explained how Ωeff​ decomposes into G-orbits in the non-transitive case.

3.Definition and interpretation of 'entropy' (Definition 2.8 & Remark 2.9): The author calls the defined quantity H(X) as 'deterministic Shannon entropy'. However, Remark 2.9 (ii) clearly states that p(x) is the 'time average access frequency'. In the case of transmission, this is indeed a uniform distribution, and the entropy is log|Ωeff|. But what does this have to do with Kolmogorov-Sinai entropy (which the authors acknowledge as 0 in Remark 3.2 (ii))? A deterministic, periodic finite system (with KS entropy 0) is described as having 'maximum uncertainty' or 'configuration complexity', rather than the 'unpredictability' of its dynamics. A simple n-periodic cycle x \to x + 1      (mod n) is described as having the maximum entropy log n, which is mathematically true, but in the context of dynamical systems, it is extremely inappropriate to equate it with the unpredictability of stochastic processes. The narrative of the thesis over-exaggerates the novelty and profundity of this result.

Response to Comment 3 (entropy interpretation overstated):

• Definition 2.8 is renamed "Stationary Shannon Entropy" and the circular phrase "induced by strong recurrence and symmetry" is removed. The measure μ is now described as "to be constructed from visit frequencies"
• Remark 3.2(ii) now clearly states the distinction with the KS entropy: “While the Kolmogorov–Sinai entropy of a deterministic permutation of a finite set is always zero (since the partition entropy of any partition grows at most polynomially), the Shannon entropy of the time-stationary distribution is non-trivial and captures the uncertainty in the state at a randomly chosen time”. This justifies the new phrase "Stationary Shannon Entropy" by the highlighting the ‘stationary’ nature of H(X).
• We state that unlike the KS entropy, H(X) measures the entropy of the time-stationary distribution on Ωeff​, which answers the question "if a time index is drawn uniformly at random from a long segment of the trajectory, how uncertain is the resulting state?" H(X) considers the distribution of states visited over time, not of the trajectory's predictability.
• We have revised the narrative and added the following to Remark 3.25(viii): the non-trivial content of Theorem 3.24 is not that the uniform distribution maximises Shannon entropy but rather that a purely deterministic system, with no randomness in its construction, is forced by its algebraic structure alone (strong recurrence and transitive G-action) to visit all states with equal long-run frequency, thereby realising the uniform distribution without any probabilistic assumption; in a generic deterministic system this fails, and the contribution of this paper is to identify the exact structural conditions under which it holds.
• We agree with the reviewer that the original manuscript overstated this result's connection to dynamical unpredictability, and the revised version is careful to frame H(X) as a measure of orbit complexity and structural coverage of state space, entirely distinct from and not in competition with the KS entropy framework.

4. There is a serious flaw in the proof of the core theorem (Theorem 3.22), Theorem 3.22 is the main result of the paper. But its proof (page 16) has fatal flaws. Unique ergodicity misuse and cyclic argument: Proof claims depend on 'unique ergodicity' and Folner sequences. However, the author assumes that the system (G, Ω, m) is ergodic and uniquely ergodic (page 15). But what is the definition of ' unique ergodicity ' when the state space Ω is a general measure space (may not be compact) and there is no topological structure? Usually, the unique ergodicity is for compact topological dynamical systems which preserve continuous transformations. More importantly, the conclusion that the author tries to prove is that 'the frequency converges to the G-invariant probability measure μ', and the proof directly uses the unique ergodicity, which is tantamount to circular argument. The goal of the theorem should be to derive the convergence of the frequency and the uniformity of the distribution from strong resilience and symmetry, rather than as a prerequisite assumption. The authors try to avoid the pointwise ergodic theorem by weak convergence and compactness argument (Remark 3.23 (iii)), but it still needs to prove that all limit points are G-invariant and all Folner sequences converge to the same limit. This is a very strong conclusion for a non-compact pure measure space system lacking topological structure, which can not be easily obtained only by strong resilience.

Response to Comment 4 (circularity and flaw in Theorem 3.22):

• A newly added Remark 3.25(i) explains why assuming unique ergodicity is not circular (it is a verifiable structural condition on the finite set Ωeff​, automatically satisfied when G acts transitively), and clarifies that what the theorem proves is the convergence of trajectory averages, not the existence of μ.
• The reviewer is correct that unique ergodicity in its classical sense is defined for compact topological dynamical systems. In the revised manuscript we have clarified that unique ergodicity is not invoked for the ambient system (G,Ω,m), which may indeed be non-compact and lack topological structure, but exclusively for the restricted system on the finite set Ωeff​. Since Ωeff is a finite set, the space P(Ωeff) of probability measures on it is a compact convex finite-dimensional simplex. Unique ergodicity in this context means simply that there is exactly one G-invariant probability measure on Ωeff ​, which is a purely measure-theoretic statement requiring no topology on Ω. This is now stated explicitly in the revised theorem and in Remark 3.25(iv).
• In the most important case, when G acts transitively on Ωeff, transitivity of G on a finite set Ωeff​ implies minimality (every G-orbit is all of Ωeff​), and minimality of a G-action on a finite set implies unique ergodicity by an elementary argument (the uniform measure is the only candidate and is manifestly G-invariant). This is now stated explicitly. Thus in the transitive case the unique ergodicity assumption is not an extra hypothesis but a consequence of transitivity, and the proof is entirely free of circularity. In the non-transitive case, unique ergodicity must be verified separately, and we provide three checkable sufficient conditions in Remark 3.25(iv).
• The reviewer asks whether the conclusion that all limit points of the empirical measures μ_n​ are G-invariant can be established for a general measure space without topology. We acknowledge that this step deserves more care, and the revised manuscript addresses it as follows. The compactness argument is applied not to the ambient space Ω but to the space P(Ωeff) of probability measures on the finite set Ωeff. This space is homeomorphic to a finite-dimensional simplex and is compact regardless of any topology on Ω. The empirical measures \mu_n = \frac{1}{|F_n|}\sum_{g \in F_n} \delta_{T_g(x_0)}​ are probability measures on Ωeff (since the trajectory stays in Ωeff​), hence live in this compact space. Every subsequential limit ν =\lim_{n_k \to \infty} \mu_{n_k}​, and the Følner property guarantees that every limit point ν is G-invariant on Ωeff ​. Unique ergodicity of the restricted system then forces ν=μ, so the full sequence converges. The role of ergodicity of the ambient system (G,Ω,m) is separate: it is used via the pointwise ergodic theorem for amenable group actions to ensure this convergence holds for m-almost every starting point x_0∈Ω, not just for a specific one. This distinction between "convergence for a.e. x_0​" and "identification of the limit as μ" is now made explicit in the proof steps and in Remark 3.23(iii)--(iv).
• The overall logical flow: (a) strong recurrence guarantees that the trajectory visits every state in Ωeff infinitely often; (b) amenability of G provides Følner sequences which turn this into convergent time averages; (c) the Følner property forces all limit points of the empirical measures to be G-invariant on the finite set Ωeff​; (d) unique ergodicity of the restricted system (automatic under transitivity, checkable in general) pins the limit down to a single measure μ; (e) ergodicity of the ambient system ensures convergence holds m-almost everywhere. None of these steps is circular, each relies only on the stated hypotheses, and the finite-dimensionality of P(Ωeff) makes the compactness argument rigorous without any topology on Ω.

5. The definition of group action on Ωeff is not clear: In the theorem statement, it is assumed that Ωeff is 'a finite effective state space invariant under G action'. However, how is the role of G on Ωeff defined? If Ωeff is generated by only one T-trajectory (a subsemigroup of G), then the action of the whole group G on Ωeff may not be well defined at all. The authors need to prove that the role of G on Ω can be limited to Ωeff and remain transitive. For non-transitive cases (e.g., Corollary 3.25), Ωeff (x_0) is more likely to be just a T-orbit. What is the G-interaction structure on it? These key issues have not been answered.

Response to Comment 5 (G-action on Ωeff​ not defined):

• The fact that the restricted action is well-defined is shown in Remark 2.3(iii) without using any commutativity condition.
• Removing T and replacing it by the whole semigroup (T_g)_{g\in G+}, now Ωeff is no longer generated by only one T-trajectory (a subsemigroup of G). This is also corrected in all subsequent results and examples.
• Both transitive and non-transitive cases now follow this corrected definition.

6. Disconnection between Abstract and text: Abstract claims that the analysis is extended to 'the role of any countable linear ordered group', but the main result of the text (Theorem 3.22) is only for 'amenable groups ', which is an important limitation and is not mentioned in the abstract. The logical break between Section 2 and Section 3: Section 2 introduces the strong resilience on general measure space, Prasad 's work and so on, but the core result of Section 3 finally settles on a finite, atomic set Ωeff. The key mathematical steps of how to transition from the former to the latter are missing. The supplementary explanation of Remark 3.27 is an ex post facto remedy, which is not enough to form a strict logical chain.

Response to Comment 6 (abstract misleading; logical gap between Sections 2 and 3):

• The abstract is fully rewritten to honestly state that recurrence results hold for arbitrary ordered groups while entropy results require amenability.
• A newly added paragraph right before Section 3.1 provides an explicit road map: results for a general group vs. results for amenable groups, and a "logical bridge" describing exactly how Theorem 3.24 connects the two levels.

Reviewer 4 Report (New Reviewer)

Comments and Suggestions for Authors

Please revise as suggested in the report.

Comments for author File: Comments.pdf

Author Response

REVIEWER 4

Response to Comment 1 on Theorem 3.8 (Uniform Bound on Minimal Recurrency Sets): The original proof was vague, citing undefined "quasi-invariance constants." The revised proof is self-contained and rigorous. It argues by contradiction: supposing minimal recurrency sets grow unboundedly, one constructs a contradictory antichain. The key insight is that in a finite-measure space, the number of essentially distinct translates of a set of measure ≥δ is bounded above by ⌊m(Ω)/δ⌋, providing the explicit uniform bound N(δ). The assumption m(Ω)<∞ is now stated clearly in the theorem, and a remark explains why it is essential.

Response to Comment 2 on ergodicity vs. unique ergodicity in Theorem 3.22 (now Theorem 3.24): The theorem statement now carefully separates the two conditions: ergodicity of (G,Ω,m) (used for pointwise a.e. convergence of Følner averages) versus unique ergodicity of the restricted system on Ωeff (used to identify the limit as μ). A remark after the result now explicitly notes that in the transitive case, transitivity implies minimality, which implies unique ergodicity automatically, so the unique ergodicity assumption is redundant there. A new Remark 3.23(iv) gives a complete logical analysis of when each condition is needed, and a new Remark before Theorem 3.25 clarifies the hypothesis requirements for the non-transitive case.

Response to Comment 3 on label inconsistency ("Corollary 3.25" vs. "Theorem 3.25"): Every occurrence of "Corollary 3.25" throughout the paper has been replaced with the uniform label Theorem 3.27.

Response to Comment 4 on applied connections: New paragraphs have been added, one in the Introduction and one in the Discussion, explicitly discussing the cited papers by Jin et al. (2024), Xiao et al. (2025), and Liu et al. (2025), explaining precisely how each relates to the paper's results. All three are added to the bibliography.

Response to Comment 5 on Følner framework used before it was introduced: The amenability assumption is not needed in Theorem 3.10 (now Theorem 3.11). Remark 3.12 after the theorem explicitly notes this.

Round 2

Reviewer 3 Report (Previous Reviewer 4)

Comments and Suggestions for Authors

The author revised the paper according to the reviewers ' comments and suggestions. The current version of this paper is acceptable for publication.

Reviewer 4 Report (New Reviewer)

Comments and Suggestions for Authors

I am satisfied with the revised version. 
The paper should be published now.
The authors should correct refrence [12] and [16].
In [12] (Jin, J.; et al.) Instead of et al., the author names should be mentioned (Liu, Q., Chen, P., Lin, K., Zhao, K., Ding, J. and Li, Y.)

In [16], instead of Liu, Q.; et al., the full author last names should be mentioned: "Jin, J., Chen, W., Ouyang, A., Yu, F., and  Liu, H. "

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript investigates discrete-time deterministic dynamical systems defined on finite state spaces, with the dynamics governed by the positive cone of countable ordered symmetry groups. The authors examine the recurrence and wandering sets with information theory. The primary contribution of this paper is showing that maximal Shannon entropy $H(X)=\log |\Omega_{eff}|$ emerges purely as a structural invariant rather than relying on stochastic or random variables. Furthermore, they demonstrate that in a discrete-time deterministic system equipped with a countable amenable ordered group acting transitively, strong recurrence eliminates weakly wandering sets, forcing all accessible states to be visited infinitely often.

Rather than restricting the analysis to classical $\mathbb{Z}$-actions, the authors extend their consideration to arbitrary countable amenable groups and Følner sequences to calculate asymptotic orbit frequencies. However, the framework specifically defines trajectories using the "positive cone $G^+$ of a countable ordered group" with the specific algebraic implications of bi-orderability barely utilized beyond providing a directed time index, leaving the necessity of the "ordered" condition questionable. It seems the discussion focuses on linealy ordered instead of generally ordered groups.

On the other hand, the finite deterministic map examples are straightforward. The paper would benefit from a more complex computational example demonstrating these constraints in a larger state space. The achievement of maximal Shannon entropy of the system relies entirely on the assumption that the symmetry group $G$ acts transitively on the effective state space. Softening the assumption would be interesting to audience.

Author Response

REVIEWER 1

Comment R1-1: Necessity of the ordered-group assumption

"The framework specifically defines trajectories using the positive cone G+ of a countable ordered group, with the specific algebraic implications of bi-orderability barely utilised beyond providing a directed time index, leaving the necessity of the 'ordered' condition questionable. It seems the discussion focuses on linearly ordered instead of generally ordered groups."

Response. The reviewer is correct on both counts. First, the results do not require bi-orderability; only the existence of a positive cone $G^+$ that forms an upward-directed semigroup is used. This is now explicitly noted in Section 2 to avoid any impression that bi-orderability is assumed.

Second, all examples ($Z, Z^2$ with lexicographic order) are linearly ordered, and the proofs use only that every pair of elements of G is comparable under the order, which is precisely linear order. We have specified this in the first paragraph of Section 2 and accordingly replaced every occurrence of 'countable ordered group' by 'countable linearly ordered group', while retaining a parenthetical remark that the proofs extend to groups whose positive cone is merely an upward-directed semigroup (no linear order required). This both answers the reviewer's concern and makes the scope of applicability wider rather than narrower.

Comment R1-2: More complex computational example

"The paper would benefit from a more complex computational example demonstrating these constraints in a larger state space."

Response. We have substantially revised Example 3.1(iii) (the $S_3$ action on $\{0,1\}^3$). We also added new instances to Example 3.1, involving in particular the case of group $Z^2$ acting on a 16-state space, illustrating the non-cyclic case promised in the abstract, with explicit orbit enumeration and entropy computation.

Comment R1-3: Softening the transitivity assumption

"The achievement of maximal Shannon entropy relies entirely on the assumption that G acts transitively on the effective state space. Softening the assumption would be interesting."

Response. We agree that this is mathematically attractive and have added a new Theorem 3.25 (non-transitive case). When the action decomposes $\Omega_{eff}$ into $k$ distinct $G$-orbits $O_1,...,O_k$, each orbit carries a uniform invariant measure (by Theorem 3.22 applied to the restricted, transitive action on each orbit), and the overall Shannon entropy is strictly less than $\log|\Omega_{eff}|$ unless all orbits have equal length (the transitive case). The corollary thus precisely quantifies the entropy deficit caused by non-transitivity and subsumes the original theorem as the $k=1$ special case.

Reviewer 2 Report

Comments and Suggestions for Authors

Referee report for the manuscript: "Recurrence and entropy for discrete-time deterministic dynamical systems"

This paper investigates discrete-time dynamical systems defined on finite state spaces endowed with symmetry groups, and further extends the analysis to the influence of countable ordered groups. The study focuses on the notion of strong recurrence, characterized by the absence of weakly wandering sets of positive measure. A theoretical framework is developed that connects classical recurrence theory (e.g., in the spirit of Hagian--Caccione) with concepts from information theory. In particular, it is shown that Shannon entropy can emerge as an intrinsic structural property of deterministic systems, without invoking any stochastic assumptions. The system is formulated via a deterministic transition function together with a countable symmetry group---often assumed to be amenable---acting transitively on the effective state space. By employing Fّlner sequences associated with amenable groups, the limiting frequencies of state visits are established, leading to the construction of a uniform invariant measure. The main result demonstrates that the entropy attains its maximal value, namely $\log |\Omega_{\mathrm{eff}}|$, as a direct consequence of the underlying algebraic structure and recurrence properties, independent of any randomness. In addition, the paper provides a detailed analysis of strong and strict recurrence, explores their connection to ergodic decomposition theory, and presents illustrative examples in both finite and infinite settings. These examples highlight, in particular, the sensitivity of maximal entropy to symmetry-breaking effects.

Despite the quality of the paper, there are some points that need improvement to ensure its acceptance:

• The paper would benefit from a clearer explanation of how an atomic probability measure is derived from an underlying non-atomic space. In particular, the construction of the effective finite space $\Omega_{\mathrm{eff}}$ should be described more precisely, including the conditions under which this reduction is valid.
• The notion of ordered groups is introduced in the definitions; however, it does not appear to play a significant role in the main results (e.g., Theorem 3.24). The authors are encouraged either to clarify the necessity of this assumption or to relax it if it is not essential for the core arguments.
• In Theorems 3.21 and 3.24, the analysis assumes the existence of a unique constant $G$. In deterministic (non-random) systems, multiple invariant constants may arise. The authors should provide a justification---either through a reference or a concise argument---that the imposed conditions ensure the uniqueness of the invariant probability measure, or at least discuss this issue explicitly.
• The relationship between the two notions introduced on pp. 10--11 requires further clarification. In particular, strict recurrence imposes a uniform lower bound condition, yet it does not seem to be used in the derivation of the main entropy result. It would be helpful to clarify whether Theorem 3.24 requires strict recurrence or whether strong recurrence alone is sufficient, and to revise the terminology accordingly for consistency.
• There are several minor typographical issues in the manuscript. For example, duplicate references appear in the bibliography (e.g., entries [5] and [10] correspond to the same work). A careful revision of the reference list is recommended to ensure consistency with the citations in the text.
• The illustrative example discussing orbit lengths (1, 3, and 4) and the resulting entropy value $\log 8$ requires further clarification. In particular, it is not clear how a uniform invariant distribution is obtained despite the presence of orbits of different lengths. The authors should explain whether these orbits are connected through group symmetry or whether the definition of $\Omega_{\mathrm{eff}}$ implicitly enforces uniformity. Alternatively, the example could be replaced with a more transparent one.
• The use of the Mean Ergodic Theorem implicitly assumes convergence with respect to an invariant measure. In a purely deterministic setting, convergence is not always guaranteed. The authors should clarify whether the conditions imposed in the paper (e.g., strong recurrence and symmetry assumptions) are sufficient to ensure convergence for all Fّlner sequences.
• The role of the effective space $\Omega_{\mathrm{eff}}$ deserves further discussion. While it is used in the definition of recurrence, entropy is computed on $\Omega_{\mathrm{eff}}$ using an atomic probability measure. It would be helpful to justify the necessity of starting from a non-atomic space rather than formulating the theory directly on the effective finite state space.
• If the group action is not fully transitive on $\Omega_{\mathrm{eff}}$, the resulting distribution may fail to be uniform. In this case, it would be interesting to investigate whether the main result can be generalized so that maximal entropy is achieved on each orbit separately, leading to a total entropy strictly below the global maximum.

I recommend acceptance of the paper after substantial revision. Addressing the points above would significantly strengthen the manuscript and enhance its clarity and impact. The work has the potential to provide a meaningful contribution by establishing a novel connection between ergodic theory and information theory in deterministic settings.

Author Response

REVIEWER 2

Comment R2-1: Construction of the atomic measure from a non-atomic space

"The paper would benefit from a clearer explanation of how an atomic probability measure is derived from an underlying non-atomic space. The construction of Omega_eff should be described more precisely."

Response. We have this in Remark 2.10(ii). The rational for starting from the non-atomic ambient space, rather than working on $\Omega_{eff}$ directly, is that the hypotheses of strong recurrence and quasi-invariance are most naturally stated for the full space $(\Omega, m)$. The finite effective space is a derived object, and the reduction to it is what allows us to apply finite combinatorics in orbit enumeration. Please note that we have relaxed the non-atomic assumption in the revised version (to accommodate certain natural examples) while specifying it when needed (like in Proposition 3.11) and addressing the significance of the non-atomic assumption to showcase the power of our methods in Remark 3.27.

Comment R2-2: Role of the ordered-group assumption in main results

Response. The short answer is that the order structure is essential in having a correct notion of recurrence. While it’s role in the proofs of the main results is implicit, it is an essential assumption in building the general theory. We have added three new paragraphs in the beginning of Section 2 to explain this and to give a literature review.

Comment R2-3: Uniqueness of the G-invariant probability measure

"In Theorems 3.21 and 3.24, the analysis assumes the existence of a unique G-invariant measure. The authors should justify that the imposed conditions ensure uniqueness."

Response. This is a genuine point. In full generality, a strongly recurrent amenable system can admit multiple invariant measures (one for each ergodic component). Theorems 3.21 and 3.24 are merged into Theorem 3.22 that carries the explicit hypothesis that the system is ergodic and uniquely ergodic.

Comment R2-4: Strict recurrence not used for entropy

"It would be helpful to clarify whether Theorem 3.24 requires strict recurrence or whether strong recurrence alone is sufficient."

Response. The reviewer is correct: strong recurrence alone was sufficient for Theorem 3.24, now Theorem 3.22 (entropy maximality), and strict recurrence plays no role there. We have added this observation to Remark 3.26 after the newly added Theorem 3.25.

 

Comment R2-5: Duplicate references [5] and [10]

Response. Corrected. References [5] and [10] were both the Eigen–Hajian–Ito–Prasad (2014) paper on minimal recurrency sets. We have retained only one entry (as [5]) and updated all in-text citations accordingly. The reference list has been carefully checked for other duplicates; none were found.

Comments R2-6 and R2-7: Example 3.1(iii) — uniform distribution despite unequal orbit lengths

Response. We revised the example with explicit T and corrected orbit lengths. In the new version all orbits have length 1 or 3 under the cyclic map T, transitivity of $S_3$ on $\{0,1\}^3$ is verified explicitly, and the uniform distribution is explicitly stated and verified.

Comment R2-8: Convergence and the Mean Ergodic Theorem in a deterministic setting

"The use of the Mean Ergodic Theorem implicitly assumes convergence with respect to an invariant measure. In a purely deterministic setting, convergence is not always guaranteed."

Response. We have clarified this point carefully. The convergence used in Theorem 3.22 of the revised version (Steps 1–3) is not the pointwise ergodic theorem (which can fail for individual trajectories) but rather the weak convergence of empirical measures along Folner sequences to the unique $G$-invariant measure. This follows from two ingredients: (a) compactness of the space of probability measures on the finite set $\Omega_{eff}$, which guarantees that every subnet of $(\mu_n)$ has a convergent subsequence; and (b) the Folner property, which implies that every limit point is $G$-invariant (as shown in Step 2 of the proof). Unique ergodicity then pins down the limit as $m$, and since the limit is unique, the full sequence also converges. This argument does not invoke the von Neumann Mean Ergodic Theorem and it is a self-contained compactness-and-uniqueness argument. We have rewritten Steps 1–3 of the proof to make this explicit. We have also added this to the Remark 3.23 after the proof of the theorem.

Comments R2-9 and R2-10: Non-transitive case and per-orbit entropy

Response. Addressed by the new Theorem 3.25 and Example 3.1(iv).

Reviewer 3 Report

Comments and Suggestions for Authors

Symmetry review

1 How do you rigorously justify the transition from a non-atomic measure space to a finite effective state space Ωeff without ambiguity?
2 Is your definition of strong recurrence equivalent to classical recurrence notions (e.g., Poincaré, Furstenberg), or strictly stronger? Provide a formal comparison theorem.
3 What additional structure on G+ (ordering, topology) is essential for the results?Would the theory still hold for non-ordered groups?
4 How sensitive are your results to the choice of Følner sequence?Is the limiting frequency unique?
5 maximal Shannon entropy emerges without randomness. Is this truly “maximal entropy” or simply uniform distribution over Ωeff How do you distinguish this from trivial combinatorial uniformity?
6 Since the original space is non-atomic, how is this atomic measure rigorously constructed?Is this a pushforward measure or a restriction?
7 deterministic system achieves stochastic maximality. In what sense is this equivalent to stochastic systems? Do you compare:entropy rate? Kolmogorov–Sinai entropy?
8 Can you provide non-trivial examples where: strong recurrence holds but system is not periodic?
9 Is strict recurrence necessary for entropy results, or is strong recurrence sufficient?
10 How is your framework fundamentally different from:deterministic finite automata permutation dynamics
11 How does this deterministic entropy mechanism translate to: noisy real-world systems? non-ideal symmetry?
12 “maximal Shannon entropy emerges without stochastic assumptions” Is this truly a non-trivial phenomenon, or simply a consequence of: finite state space, transitive group action
13  Authors may find useful to refer 10.1016/j.chaos.2026.117913 which discusses robust chaotic properties of 1d ceil map
14 Which step of Theorem 3.24 fails for non-amenable groups?
15 Can you provide: non-trivial examples beyond cyclic shifts and rotations? systems that are not periodic but strongly recurrent?
16 If orbits are not equal, how do you get uniform distribution over all 8 states?Does transitivity fail here?
17 The construction of a weakly wandering set from sparse returns is non-trivial. a rigorous construction or cite a theorem?
18 You invoke the Mean Ergodic Theorem implicitly.measure-preserving?ergodic action?
19 How do you rigorously justify treating a deterministic trajectory as a random variable without invoking: ergodicity assumptions probabilistic frameworks

Author Response

REVIEWER 3

Comment R3-1: Transition from non-atomic to finite effective state space

Response. We have added a dedicated Remark 2.9(ii) of how to handle this under unique ergodicity for amenable case (see Theorem 2.22 in the revised version). The rational for starting from the non-atomic ambient space, rather than working on $\Omega_{eff}$ directly, is that the hypotheses of strong recurrence and quasi-invariance are most naturally stated for the full space $(\Omega, m)$. The finite effective space is a derived object, and the reduction to it is what allows us to apply finite combinatorics (orbit enumeration, Theorem 3.22 plus new Example 3.24). Please note that we have relaxed the non-atomic assumption in the revised version (to accommodate certain natural examples) while specifying it when needed (like in Proposition 3.11) and addressing the significance of the non-atomic assumption to showcase the power of our methods in Remark 3.27.

 

Comment R3-2: Formal comparison of strong recurrence with Poincare and Furstenberg recurrence

"Is your definition of strong recurrence equivalent to classical recurrence notions, or strictly stronger? Provide a formal comparison theorem."

Response. We have added this to the second paragraph after Remark 2.6. This is almost immediate, so we added that as a comment in the paragraph not a separate result.

Comment R3-3: What structure on G+ is essential

Response. No topology and no bi-orderability are required. The linear-order assumption is added, and some remarks are added about bi-orderability.

Comment R3-4: Sensitivity to the choice of Folner sequence and uniqueness of limiting frequency

"How sensitive are your results to the choice of Folner sequence? Is the limiting frequency unique?"

Response. Under unique ergodicity, the limiting frequency is the same for every Folner sequence and for $m$-almost every initial condition; it equals $p(x) = \mu(\{x\}) = 1/|\Omega_{eff}|$ in the transitive case. Without unique ergodicity, different Folner sequences or initial conditions can give different limits (one per ergodic component). We have made this explicit in Remark 3.23(iv).

Comment R3-5: The 'maximal entropy' versus uniform distribution over Omega_eff?

"Is this truly 'maximal entropy' or simply uniform distribution over Omega_eff? How do you distinguish this from trivial combinatorial uniformity?"

Response. This is a valid conceptual point. The claim is not that the uniform distribution is surprising in itself. The uniform distribution over $n$ states maximises Shannon entropy among all distributions on $n$ states, by the standard concavity argument. The non-trivial content is that a purely deterministic, non-random system is forced by its algebraic structure (strong recurrence plus transitive symmetry) to visit every state with equal long-run frequency, thereby realising the uniform distribution without any probabilistic assumption. In a generic deterministic system this is false: the system might be periodic with period shorter than $|\Omega_{eff}|$ or might visit some states more often than others. Our contribution is to identify the exact structural conditions (strong recurrence and transitive $G$-action) that are jointly necessary and sufficient to force uniformity. We have added Remark 3.2(i) to clarify this point.

Comment R3-7: Comparison with stochastic systems — entropy rate vs. KS entropy

"In what sense is the deterministic entropy equivalent to stochastic systems? Do you compare entropy rate or Kolmogorov–Sinai entropy?"

Response. The entropy $H(X)$ is the Shannon entropy of the stationary distribution on the state space, not the Kolmogorov–Sinai (metric) entropy of the dynamical system (which would be zero for a periodic deterministic system). We have added a Remark 3.2(ii) that carefully distinguishes these two notions: (a) the KS entropy of a deterministic permutation of a finite set is always zero, because the map is bijective and the partition entropy of any partition grows at most polynomially; (b) our $H(X)$ is the Shannon entropy of the time-stationary distribution, which measures the uncertainty in the state at a randomly chosen time, not the complexity of the orbit structure. The analogy with stochastic systems is that a uniformly ergodic Markov chain on $n$ states also has stationary Shannon entropy $\log n$, whereas our result shows a deterministic system can achieve the same time-averaged uncertainty.

Comment R3-8: Non-trivial examples where strong recurrence holds but system is not periodic

Response. Example 3.1(vi) (Sturmian shift) is added along with some other non-trivial examples.

 

Comment R3-9: Is strict recurrence necessary for entropy results?

Response. No. Strong recurrence alone is sufficient for Theorem 3.24, now Theorem 3.23 (entropy maximality), and strict recurrence plays no role there. We have added this observation to Remark 3.26 after the newly added Theorem 3.25.

Comment R3-12: Is maximal entropy a non-trivial phenomenon?

Response. Yes, see the response to Comment R3-5.

Comment R3-13: New reference 10.1016/j.chaos.2026.117913

Response. We thank the reviewer for bringing this reference to our attention. The paper (on robust chaotic properties of the one-dimensional ceil map) is relevant to our discussion of deterministic systems achieving stochastic-like complexity. We have added it to the bibliography as [chaos2026] and cited it in the Introduction [15].

Comment R3-14: Which step of Theorem 3.23 fails for non-amenable groups?

"Which step of Theorem 3.24 fails for non-amenable groups?"

Response. Step 2 of the proof ($G$-invariance of the limit) fails. For a non-amenable group, no Folner sequence exists; the Folner condition $||gF_n Delta F_n|| / |F_n| \to 0$ is precisely the definition of amenability. Without it, one cannot conclude that a limit point of the empirical measures $\mu_n$ is $G$-invariant, because the symmetrisation argument in Step 2 breaks down. We have added Remark 3.23(i) that pinpoints this failure and notes that for non-amenable groups the entropy result is an open problem.

Comment R3-17: Rigorous construction of a weakly wandering set from sparse returns

Response. We have replaced the informal Steps 2–3 with a rigorous inductive construction.

Comment R3-18: Mean Ergodic Theorem — measure-preserving and ergodic action

Response. We have clarified this point carefully. The convergence used in Theorem 3.22 in the revised version (Steps 1–3) is not the pointwise ergodic theorem (which can fail for individual trajectories) but rather the weak convergence of empirical measures along Folner sequences to the unique $G$-invariant measure. This follows from two ingredients: (a) compactness of the space of probability measures on the finite set $\Omega_{eff}$, which guarantees that every subnet of $(\mu_n)$ has a convergent subsequence; and (b) the Folner property, which implies that every limit point is $G$-invariant (as shown in Step 2 of the proof). Unique ergodicity then pins down the limit as $m$, and since the limit is unique, the full sequence also converges. This argument does not invoke the von Neumann Mean Ergodic Theorem and it is a self-contained compactness-and-uniqueness argument. We have rewritten Steps 1–3 of the proof to make this explicit. We have also added this to the Remark 3.23 after the theorem.

Comment R3-19: Justification for treating a deterministic trajectory as a random variable

"How do you rigorously justify treating a deterministic trajectory as a random variable without invoking ergodicity assumptions?"

Response. The random variable $X$ in Definition 2.1 is not the trajectory itself but a random variable whose distribution equals the time-average frequency of state visits along the trajectory. This is analogous to the way one defines an empirical distribution from a deterministic sequence: given a sequence $(x_1, x_2, \cdots, x_n)$, the empirical measure $(1/n) \sum_i delta_{x_i}$ is a deterministic object, but it can be regarded as the distribution of a random variable that selects a time uniformly at random. The 'randomness' is entirely in the time-selection, not in the dynamics. Under unique ergodicity this empirical measure converges to the unique invariant measure mu, and X is then defined to have distribution mu. No ergodicity assumption beyond unique ergodicity is required. We have added a paragraph to the Remark 2.2(ii) explaining this time-averaging interpretation.

Reviewer 4 Report

Comments and Suggestions for Authors

In this paper, we try to give a unified theoretical characterization of the strong resilience, group action symmetry and Shannon entropy of deterministic dynamical systems on finite state space, and pay special attention to how the strong resilience and transitive symmetry force the system to reach the maximum entropy under the action of amenable groups. This direction has theoretical depth, involving the intersection of ergodic theory, entropy theory and algebraic dynamical systems. However, this article has some problems in the current version and needs to be further modified : 1. The definition of 'strong resilience 'is inconsistent with the standard term. In Definition2.4, we say that ( G, Ω, m ) is strongly resilient. If for any m ( B ) > 0, there exists a finite F ∂ G + such that for any t ∈ G, there exists s ∈ F such that m ( tB ∩ sB ) > 0. This definition is not directly equivalent to the classical ' strong recurrence ' ( strong recurrence, which often means that each positive test set intersects itself under a nontrivial group element ) or ' no weak wandering set ' in ergodic theory. In Remark 3.3, the author claimed that ' strong resilience is equivalent to no weak wandering set ', but did not provide a proof, and this equivalence is far from obvious under general non-invariant measures. The definition requires that for all t ∈ G rather than only for positive time, it is easy to cause confusion in the context of ordered group. 2. The definition of ' strictly recurrent ' lacks motivation. The unified lower bound of ε > 0 is introduced in Definition 2.5, but the concept is not systematically used in the subsequent, only redescribed as syndetic property in Theorem 3.22. In the proof of Theorem 3.22, the ε defined by Step 3 depends on B and s, rather than being truly unified in all B, which is inconsistent with the definition of 2.5, ' there is ε > 0 for all B '. 3.Effective state space Ωeff is fuzzy, Ωeff = { x ∈ Ω : t, Xt = x } is defined in 2.1. However, Xt depends on the initial condition, so Ωeff also depends on the initial state. However, it is unreasonable to regard | Ωeff | as a global constant of the system in the following ( such as Theorem 3.24 ). 4. The inverse propositional error of Theorem 3.5, Theorem 3.5 states that if ( G, Ω, m ) is conservative and has an equivalent finite invariant measure, then it is strongly recurrent. In ergodic theory, conservative + finite invariant measure usually means ' recurrent ', but not necessarily the ' strong resilience ' defined by the author. For example, a ergodic measure-preserving system may not satisfy the finiteness condition ' there exists s ∈ F for all t ' for some sets. The author does not give a derivation from the finite invariant measure to its definition 2.4, but only vaguely says ' implies strong recurrence under ν ', which is a circular argument. 5. The proof of Theorem 3.11 does not hold, and Step 2 says ' If RB has zero density, then a weak wandering set can be constructed '. However, the weak wandering set requires a sequence of disjoint translations, and the density of zero is only a necessary condition rather than a sufficient condition. The author does not give a construction method. The application of ' pigeonhole principle ' on infinite sets is not properly defined. 6. The core conclusion of Theorem 3.21 is not proved, which claims that the asymptotic frequency of each G-orbit being accessed is equal to | O | / | Ωeff | in a finite state space. In the proof, it is claimed that ' strong resilience + finite state space → existence of invariant measure ', but it is not proved that the measure is unique, and it is not proved that Folner converges to the measure on average. More importantly, in the deterministic dynamical system Xt + 1 = T ( Xt ), the orbit is uniquely determined by the initial state. Unless the system is transitive and the G action is transferable, different initial states may lead to different orbit distributions. The author implicitly assumes that ' all initial states have the same asymptotic distribution ', which is not derived from strong resilience or symmetry. 7.Wrong attribution of Theorem 3.24, which claims that if G is transitive on Ωeff, then H ( X ) = log | Ωeff |. However, the definition of Shannon entropy H ( X ) depends on the probability distribution p ( x ), which is defined as the Folner mean. In deterministic systems, p ( x ) is generally not equal unless the system is periodic and uniformly accesses all states. In Step 2, the author said that ' p ( gx ) = p ( x ) is obtained by G-invariance ', but p ( x ) is defined by the trajectory, and the trajectory does not necessarily satisfy p ( gx ) = p ( x ) unless the G-action exchanges with the dynamics and the trajectory itself is G-invariant - which is generally not true in deterministic systems. In fact, the author confuses ' the measure μ is G-invariant ' with ' the asymptotic distribution defined by a single trajectory is G-invariant '. The latter requires strong conditions such as ergodicity + unique ergodicity. 8.Example 3.1 ( iii ) is not verified, and it is claimed that the S3 action on Ω = { 0,1 } 3 gives three orbital lengths of 1,3,4, and reaches the maximum entropy log8. But how does the track with a length of 4 achieve uniform distribution in 8 states ? The transfer mapping T is not provided and cannot be verified. 9.Literature citation is insufficient. ' our recent paper [ 2 ] ' is mentioned many times in the article, but the specific information of [ 2 ] is not listed in the reference ( the list of references is missing ). In addition, for the relationship between Folner sequences and entropy on amenable groups, the classical literature such as Ollagnier 2002, Kerr & Li 2016 should be cited, but it is not found in this paper. Recommendations : 1.Re-examine the definition of ' strong resilience ', either using the existing standard definitions in the field ( such as no weak wandering set ), or explicitly proving the equivalence between the new definition and the old definition. 2.Provide a complete and strict proof for each main theorem, and avoid relying on fuzzy expressions such as 'obvious ', ' available from... '. 3.Supplement missing references and correct typesetting and symbol errors. 4.Consider narrowing the scope to more specific cases ( such as finite groups, cyclic groups ), first establish a solid small result, and then try to promote.

Author Response

REVIEWER 4

Comment R4-1: Definition of strong recurrence vs. classical 'no weakly wandering set' and inconsistency with the standard terminology.

"The definition of 'strong recurrence' is not directly equivalent to classical strong recurrence or 'no weakly wandering set'. The equivalence in Remark 3.3 is claimed without proof, and this equivalence is far from obvious under general non-invariant measures."

Response. Everywhere instead of ‘'strong resilience’ we use ‘'strong recurrence’. The reviewer is right that this equivalence is non-trivial and deserved a more careful treatment. The equivalence with 'absence of weakly wandering sets of positive measure' is now moved and commented in the first paragraph after Remark 2.6, saying how it follows from slight modification of the proof of Theorem 3.2 of [2]. Regarding the possible confusion about 'for all t in G vs. for positive time only', we added Remark 2.6.

Comment R4-2: Motivation and use of strict recurrence

Response. The reviewer is correct: strong recurrence alone is sufficient for Theorem 3.24, now Theorem 3.22 (entropy maximality), and strict recurrence plays no role there. We have added this observation to Remark 3.26 after the newly added Theorem 3.25.

Comment R4-3: Omega_eff depends on initial condition

Response. Addressed in Remark 2.2(i).

Comment R4-4: Converse direction of Theorem 3.4

"The converse (conservative + finite invariant measure => strongly recurrent) is circular: the author says 'implies strong recurrence under nu' without deriving it from Definition 2.4."

Response. The reviewer is correct that the converse direction needs a cleaner argument. The converse direction in Theorem 3.4 (now called Proposition 3.4) is corrected and expanded.

Comment R4-5: Step 3 of Theorem 3.10(ii) — wandering-set construction from sparse returns

"Step 2 says 'if R_B has zero density, then a weakly wandering set can be constructed', but zero density is only a necessary condition, not sufficient for the construction. The application of the pigeonhole principle on infinite sets is not properly defined."

Response. The reviewer identifies a gap. We have replaced the informal Steps 2–3 with a rigorous inductive construction.

Comment R4-6: Theorem 3.20 — uniqueness of invariant measure and initial-condition dependence

Response. Addressed the initial condition dependence in Remark 2.2(i). Theorem 3.20 of the original version is now merged into Theorem 3.22 as part (i). Part (ii) of Theorem 3.22 in the revised version replaces Theorem 2.23 on Entropy in the original version. This was needed as the proof of these theorems in the original version were using the same ideas.  

Comment R4-7: Theorem 3.23, Step 5 — G-invariance of p(x) from G-invariance of mu

"The author confuses 'the measure mu is G-invariant' with 'the asymptotic distribution defined by a single trajectory is G-invariant'. The latter requires strong conditions such as ergodicity + unique ergodicity."

Response. The reviewer correctly identifies the subtle point and needed conditions are now added to the result (now Theorem 2.22(ii)). In the revised proof, the argument in defines $p(x)$ not as the limiting frequency along a single deterministic trajectory from a fixed initial condition but as $p(x)=\mu(\{x\})$, where $\mu$ is the unique $G$-invariant probability measure on $\Omega_{eff}$ (whose existence and uniqueness are now explicit hypotheses). The $G$-invariance $p(gx) = p(x)$ then follows immediately from the $G$-invariance of $m$, with no reference to any individual trajectory. The connection between $\mu$ and single-trajectory frequencies is then a consequence, not a premise: under unique ergodicity, the Folner averages along $m$-almost every trajectory converge to $m$. We have rewritten the proof to separate these two levels clearly. We also noted this in Remark 3.23(ii).  

Comment R4-8: Example 3.1(iii) not verified

Response. We have substantially revised Example 3.1(iii) (the $S_3$ action on $\{0,1\}^3$).

Comment R4-9: Missing reference [2] and insufficient citations

"'Our recent paper [2]' is mentioned many times but [2] is not listed in the references."

Response. Reference [2] (Amini–Swid 2023, RAIRO) is listed in the bibliography. Additionally, we have added in-text citations for some other non-cited references.