{"id":"ae5da649-70e2-4608-8237-b3a6954d99ec","arxiv_id":"1908.02930","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Zero entropy symbolic shifts always admit a probability measure invariant under every automorphism of the shift.","lead":"Every zero-entropy symbolic shift, a kind of infinite sequence space, is shown to have a probability measure that stays unchanged under all its symmetries. The paper also proves the symmetry groups of minimal such shifts are sofic, a concrete step toward an open conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4's proof skips a non-obvious subsequence lemma; if the lemma is false the main theorem collapses, but it appears provable for monotone L.","rationale":"The reader identified the correct weakest point: the passage from liminf L(r)/r = 0 to a subsequence with vanishing local increments is asserted without proof, and the entire existence theorem depends on it. My own analysis suggests the subsequence lemma is true for nondecreasing integer-valued log-growth functions, because sublinear growth forces long flat stretches between jumps, and one can choose r_n in such a stretch while keeping L(r_n)/r_n small. The paper's claim is therefore likely correct, but the proof as written contains a genuine gap: a referee should require the missing argument. I also noticed the left/right multiplication mismatch in Proposition 2.1; this is a real inconsistency in the statement, but since Theorem 1.4 uses balls, it does not threaten the main theorem and is easily repaired. For these reasons I do not think the verdict should move: the paper should be accepted conditionally on filling in the subsequence lemma and correcting the multiplication convention. This is the same conclusion as the reader's, so no change to the reader's verdict is needed.","tokens_in":6023,"tokens_out":27939,"duration_ms":309007,"concrete_test":"Prove or disprove the subsequence lemma: for every nondecreasing integer-valued L with liminf_r L(r)/r = 0, there exists r_n -> infinity such that L(r_n+i) - L(r_n) -> 0 for every fixed i. If the lemma is false, exhibit an explicit L satisfying the liminf condition but having no such subsequence, and check whether L = log N_Sigma(B_r) can occur for a subshift. If the lemma is true, fill in the missing argument in the proof of Theorem 1.4 and, separately, repair Proposition 2.1 by aligning the left- and right-multiplication conventions or by stating the condition for both.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is in the proof of Theorem 1.4: from liminf_r L(r)/r = 0 one asserts, 'because L(r) is increasing', that a subsequence r_n exists with L(r_n+i) - L(r_n) -> 0 for each fixed i. This is a nontrivial local-vanishing lemma, not an immediate consequence of monotonicity, and it is exactly what is needed to verify the hypothesis of Proposition 2.1. If it failed, the construction would not produce a characteristic measure. I believe the lemma is true for nondecreasing integer-valued L: when L(r)/r has arbitrarily small values, the jumps in L cannot occur too often, and one can choose r_n just after a large jump so that the next several increments are zero; however, the paper supplies no proof of this. A secondary issue is that Proposition 2.1's hypothesis uses left multiplication, union_{g in K} g F_n, while the proof defines F~_n = union_{g in K} F_n g, which in a nonabelian group need not be the same set. For the ball subsequence used in Theorem 1.4 both sets lie in B_{r_n + |K|}, so this is repairable, but the proposition as stated is not proved as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies characteristic measures for symbolic dynamical systems, i.e., probability measures invariant under the automorphism group of a shift. The main result, Theorem 1.4, states that for any finitely generated group G, every shift (G,Σ) with liminf_r (1/r) log NΣ(B_r) = 0 admits a characteristic measure; Theorem 1.2 is the Z specialization to zero entropy shifts. The paper also proves Theorem 1.7: the automorphism group of a minimal zero entropy Z-shift is sofic, via a more general criterion (Theorem 2.2). The proofs are based on a construction (Proposition 2.1) that produces a characteristic measure as a weak limit of uniform measures on sets of representatives of restrictions to a growing sequence of finite sets, provided that the growth is asymptotically unchanged by translating by finite sets.","tokens_in":6249,"tokens_out":7757,"duration_ms":88976,"significance":"If the technical gaps are repaired, the results are significant: they give the first general existence theorem for characteristic measures on zero entropy shifts, and the soficity conclusion is a concrete partial step toward the Cyr–Kra conjecture on amenability of automorphism groups of minimal zero entropy shifts. The central construction is elegant and self-contained, uses no fitted parameters, and does not rely on the conjectures it discusses. The paper is likely to be of interest to researchers in symbolic dynamics and topological dynamics.","major_comments":[{"comment":"The step reading 'Thus, and because L(r) is increasing, there is another subsequence r_n such that for every i>0, lim_L(r_n+i)-L(r_n)=0' is not justified in the text. This local-vanishing property is exactly what is needed to verify the hypothesis of Proposition 2.1, so the proof of the main theorem is incomplete as written. The statement is, in fact, provable for monotone integer-valued L: if for some fixed i there were a bound R such that every interval of length i beyond R contained an increment of L, then L(r) ≥ (r-R)/i for large r, contradicting liminf L(r)/r = 0; hence arbitrarily large i-increment-free intervals exist, and a diagonal choice gives the desired subsequence. This argument should be included.","section":"Proof of Theorem 1.4"},{"comment":"There is a mismatch between the hypothesis and the proof concerning left versus right multiplication. The hypothesis asserts liminf_n NΣ(∪_{g∈K} gF_n)/NΣ(F_n)=1, but the proof defines F~_n = ∪_{g∈K} F_n g and uses the inclusion gK ⊆ F_n K to show that φ' is well defined. In a nonabelian group these sets are generally different, and the proof as written therefore does not prove the proposition as stated. The same issue appears in Theorem 2.2. The applications to balls in Theorem 1.4 are unaffected because B_r K and K B_r are both contained in B_{r+diam K}, but the statements of Proposition 2.1 and Theorem 2.2 should be corrected (preferably to right multiplication) or the proof should be adapted accordingly.","section":"Proposition 2.1"},{"comment":"The claim that the fourth condition of Lemma 2.3 follows from 'the fact that K ⊆ F~' is not correct as written: K is not necessarily contained in F~ = ∪_{g∈K} gF_k. To make the argument valid, one should choose k large enough that F_k contains K (possible because the increasing sequence F_n exhausts G), so that σ_F = φ(σ)_F implies σ_K = φ(σ)_K and the defining property of K applies. This is a repairable gap, but it should be fixed explicitly.","section":"Proof of Theorem 2.2, property (4)"}],"minor_comments":[{"comment":"There is a typo in the abstract: 'adm it' should be 'admit'.","section":"Abstract"},{"comment":"In condition (4) of Lemma 2.3, the expression '~g(a) = ( a)' appears to be a typo for '~g(a) = a'.","section":"Lemma 2.3"},{"comment":"Theorem 1.7 has a grammatical typo: 'Let (Z, Σ) a minimal shift' should be 'Let (Z, Σ) be a minimal shift'.","section":"Theorem 1.7"},{"comment":"In the proof of Proposition 2.1, the phrase 'Since their union is contained in Σ_{F~_n}' is slightly imprecise: the union of R_n and S~_n is a subset of the set of projections Σ_{F~_n}, not of the full shift-invariant set Σ. The intended inequality is clear, but the wording could be tightened.","section":"Proof of Proposition 2.1"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a novel and elegant construction, and the main theorems appear correct after the indicated repairs. The two principal gaps—the unproved subsequence lemma in Theorem 1.4 and the left/right multiplication mismatch in Proposition 2.1—are localized and repairable, so I do not recommend rejection. I would ask the authors to supply the missing lemma and to correct the statements/proofs of Proposition 2.1 and Theorem 2.2 before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real result. The paper shows every Z-shift of zero entropy has a characteristic measure, generalizes to finitely generated groups with sublinear log-growth of balls, and proves soficity of Aut for minimal zero entropy shifts. The Cyr-Kra conjecture is context, not an input. I read the construction carefully; the weak-limit argument in Prop 2.1 is sound and the total variation bound is right.\n\nWhat is new: existence for zero entropy shifts answers a natural open question, the growth criterion is new, and the soficity theorem is a concrete step toward the Cyr-Kra amenability conjecture. No fitted parameters, no self-citation tricks, and the citation pattern is clean. The paper is honest about what remains open (Question 1.3).\n\nSoft spots, in order:\n\n1. The proof of Thm 1.4 skips a lemma. From liminf L(r)/r = 0 for a nondecreasing L, it claims there is a subsequence r_n with L(r_n+i) - L(r_n) -> 0 for each fixed i. That is not immediate from monotonicity, and it is exactly what feeds Prop 2.1. I believe the lemma is true -- choose r_n in a long flat stretch before a jump -- but the paper should prove it. Without that, Thm 1.4 has a gap, albeit a fillable one.\n\n2. Prop 2.1 has a left/right multiplication mismatch. The statement assumes liminf N_Sigma(union_{g in K} g F_n)/N_Sigma(F_n) = 1, but the proof uses tilde F_n = union_{g in K} F_n g. In a nonabelian group those are different sets. For the ball application both are contained in B_{r_n + diam K}, so the proof of Thm 1.4 can be repaired; as stated, the proposition is not proved. This is a presentation bug, not a conceptual one.\n\nAlso minor: the phrase \"and because L(r) is increasing\" is doing too much work. Give the subsequence argument in a sentence or refer to a lemma.\n\nOverall, the main theorem is true as far as I can tell, and the flaws are technical exposition issues. The paper deserves a serious referee; I would send it out. It will be useful to people working on automorphism groups of shifts, and to anyone following the Cyr-Kra conjecture. I would cite it if I worked in the area.","headline":"A solid, mostly self-contained proof that zero entropy shifts admit characteristic measures, with two technical gaps in the write-up that are repairable rather than fatal.","tokens_in":6759,"tokens_out":5019,"would_cite":true,"duration_ms":58110,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B10","37B05","37A35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Zero-entropy shifts always admit a probability measure invariant under every automorphism of the system.","keywords":["characteristic measure","zero entropy shift","automorphism group","symbolic dynamics","sofic group","subexponential growth","minimal shift","invariant measure"],"falsifier":"Exhibit a zero-entropy shift whose automorphism group has no invariant probability measure, or a minimal zero-entropy shift whose automorphism group is not sofic; either example would directly refute Theorems 1.2 and 1.7. A more local check is to compute ratios $N_\\Sigma(B_{r+R})/N_\\Sigma(B_r)$ for fixed $R$: a zero-entropy shift for which these ratios stay bounded away from 1 along every subsequence would break the subsequence step of Theorem 1.4.","tokens_in":5808,"feed_emoji":"🎲","tokens_out":11509,"duration_ms":114739,"temperature":0.7,"pith_summary":"This paper proves that every symbolic dynamical system with zero topological entropy carries a characteristic measure: a Borel probability measure fixed by all automorphisms of the system. This matters because automorphism groups of shifts are often large and non-amenable, so such a measure is not guaranteed by general fixed-point theorems. The proof uses the shift's growth function: when the number of allowed patterns on radius-$r$ balls grows subexponentially, uniform measures on restrictions of configurations to carefully chosen balls converge to a limit that every automorphism preserves. The same machinery also shows that automorphism groups of minimal zero-entropy shifts are sofic, a property weaker than amenability.","feed_headline":"Zero-entropy shifts always have a symmetry-invariant measure","feed_subtitle":"A measure fixed by every automorphism exists whenever the shift's word-count growth is subexponential.","key_machinery":"The central object is the growth function $N_\\Sigma(F)$, the number of distinct finite patterns the shift allows on a finite set $F$. The engine is a ratio condition: there must be an increasing exhaustive sequence of finite sets $F_n$ with $N_\\Sigma(\\bigcup_{g\\in K} gF_n)/N_\\Sigma(F_n)\\to 1$ for every finite $K$, and zero entropy guarantees this for balls. Proposition 2.1 shows that under this condition, the uniform measures on sets of representatives of the restrictions to $F_n$ converge along a subsequence to a characteristic measure. Automorphisms enter through their memory sets—the finite sets on which the automorphism acts as a fixed block map—so that on each $F_n$ an automorphism is approximately a bijection between representative sets, forcing the limit measure to be invariant.","core_discovery":"The central theorem (Theorem 1.4) states that for every finitely generated group $G$ and every shift $(G,\\Sigma)$ with $\\liminf_{r\\to\\infty} \\frac{1}{r}\\log N_\\Sigma(B_r)=0$, the shift admits a characteristic measure; specializing to $G=\\mathbb{Z}$ gives Theorem 1.2, that every zero-entropy shift over the integers admits one. The proof constructs the measure as a weak limit of uniform measures on sets of representatives of restriction maps $\\pi_n:\\Sigma\\to\\Sigma^{F_n}$, and shows that any automorphism, being a block map with finite memory, pushes the limit back to itself. A parallel argument establishes that $\\mathrm{Aut}(\\mathbb{Z},\\Sigma)$ is sofic whenever $(\\mathbb{Z},\\Sigma)$ is a minimal zero-entropy shift. The broader question of whether every shift, regardless of entropy, admits a characteristic measure is left open.","pith_inferences":["The ratio condition in Proposition 2.1 looks like the real sufficient hypothesis, and it does not require the group to be amenable; finding other sequences of sets with negligible relative boundary growth might yield characteristic measures outside the zero-entropy setting.","The soficity result is a step toward the open conjecture that automorphism groups of minimal zero-entropy shifts are amenable; showing that the constructed characteristic measure arises from an amenable action would close the remaining gap.","Lemma 2.4 suggests that in minimal shifts every nontrivial automorphism moves every configuration, and it is exactly this fixed-point-free behavior that converts the growth condition into soficity; non-minimal shifts would need a different mechanism.","A natural computational check on any proposed zero-entropy shift is whether the uniform representative measures converge in the sense of Proposition 2.1; shifts where the limits depend on the chosen subsequence would reveal extra structure in the automorphism group."],"forward_implications":["Every zero-entropy shift over $\\mathbb{Z}$ has a probability measure invariant under all its automorphisms.","Every shift over a finitely generated group whose ball-pattern count grows subexponentially admits a characteristic measure; for amenable groups the measure can be chosen invariant under the group action as well.","Every minimal zero-entropy shift has a sofic automorphism group, placing these groups in a broad class that contains amenable and residually finite groups.","The construction is explicit enough that characteristic measures can be approached as limits of uniform cylinder measures for concrete shifts."],"supporting_citations":[{"why":"Supplies the memory-set representation of automorphisms as block maps with finite memory, the device that lets the proof compare an automorphism's pushforward with the limit measure on finite sets.","marker":"[2]"},{"why":"Establishes that automorphism groups of shifts are countable, the structural fact that lets finite subsets of Aut be used in the soficity argument.","marker":"[11]"},{"why":"Provides the working definition of sofic group used in Lemma 2.3, so the partial-action construction in Theorem 2.2 yields soficity of Aut.","marker":"[12]"}],"fun_headline_variants":["Zero entropy guarantees a symmetry-fixed measure","Symmetry-invariant measures always exist for zero entropy shifts","Zero entropy shifts: characteristic measures always exist","Minimal zero entropy shifts have sofic automorphism groups","Zero entropy means automorphism-invariant measure exists"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that zero exponential growth lets you pick radii at which enlarging a ball by any fixed finite set adds only a negligible fraction of new allowed patterns; if this subsequence does not exist, the constructed limit measure need not be automorphism-invariant.","fun_headline_variants_meta":{"raw":{"variants":["Zero entropy guarantees a symmetry-fixed measure","Symmetry-invariant measures always exist for zero entropy shifts","Zero entropy shifts: characteristic measures always exist","Minimal zero entropy shifts have sofic automorphism groups","Zero entropy means automorphism-invariant measure exists"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00061,"raw_usage":{"total_tokens":2744,"prompt_tokens":752,"completion_tokens":1992,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":368,"completion_tokens_details":{"reasoning_tokens":1919}},"tokens_in":368,"tokens_out":1992,"duration_ms":15266,"temperature":1.0,"reasoning_tokens":1919,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:31:27.558806+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a zero-entropy shift whose automorphism group has no invariant probability measure, or a minimal zero-entropy shift whose automorphism group is not sofic; either example would directly refute Theorems 1.2 and 1.7. A more local check is to compute ratios $N_\\Sigma(B_{r+R})/N_\\Sigma(B_r)$ for fixed $R$: a zero-entropy shift for which these ratios stay bounded away from 1 along every subsequence would break the subsequence step of Theorem 1.4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the memory-set representation of automorphisms as block maps with finite memory, the device that lets the proof compare an automorphism's pushforward with the limit measure on finite sets."},{"cited_title":"Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Mathematical systems theory 3 (1969), no","cited_arxiv_id":null,"evidence_quote":"Establishes that automorphism groups of shifts are countable, the structural fact that lets finite subsets of Aut be used in the soficity argument."},{"cited_title":"https://web.ma.utexas.edu/users/juschenko/files/soficgroups.pdf","cited_arxiv_id":null,"evidence_quote":"Provides the working definition of sofic group used in Lemma 2.3, so the partial-action construction in Theorem 2.2 yields soficity of Aut."}],"review_version":1}