Cпектры cамоподобных эргодических действий

The paper states and proves two claims: (i) for any M⊂N there exists a Poisson suspension with spectral multiplicity set M∪{∞} (Theorem 1), and (ii) there exists a Gaussian flow St such that the time-maps Sp^n have multiplicities {1,∞} for n≤0 and {p^n,∞} for n>0, and an associated infinite tensor-power flow Tt whose T_{p^n} are pairwise non-isomorphic for n≤0 and pairwise isomorphic for n>0 (Theorem 2). Both claims, together with the key functorial identity that G(T) and P(T) are spectrally isomorphic to exp(T)=⊕_{n≥0}T^{⊙n}, are explicitly stated and used in the note (see the abstract and statements of Theorems 1 and 2, and the exp-functor identity in the introduction: G(T),P(T)≅exp(T) ). The proof of Theorem 1 builds specific self-similar rank-one transformations T_m of type (p_m,8) and shows that exp(⊕_{m∈M} mT_m) has multiplicity set M∪{∞}, with higher chaoses yielding a countable (i.e., infinite) multiplicity and being spectrally disjoint from the first chaos (see the decomposition with Q and the intertwining argument) . Theorem 2 is obtained by constructing an orthogonal flow Ot with the p-self-similar spectral structure so that Op^n has simple spectrum for n≤0 and homogeneous multiplicity p^n for n>0, and then passing to the Gaussian flow St=exp(Ot); the tensor-power variant Tt is then handled explicitly . By contrast, the model’s “Phase 2” solution contains a fatal error in the flow part: starting from O1≅⊕_{j=0}^{p−1}e^{2πij/p}U, one has Op^n=O1^{p^n}=⊕_{j=0}^{p−1}U^{p^n}, giving at most multiplicity p (not p^n) on the first chaos. The claimed growth to p^n does not follow from that construction. The automorphism/flow claims for negative times are also left unjustified. While the model’s method for Poisson multiplicities M∪{∞} is broadly in the right spirit (via exp-functor and disjoint rotated copies), several steps are under-justified (e.g., ensuring the needed rotated phases via group extensions; ensuring disjointness; and the precise route to ∞ via higher chaoses). Hence the paper’s results and proofs are consistent and correct as presented, whereas the model’s proof, particularly for the flow, is incorrect.