ON BOUNDED COMPOSITION OPERATORS ON FUNCTIONAL QUASI-BANACH SPACES AND A STABILITY OF DYNAMICAL SYSTEMS

The paper’s Theorem 2.4 proves |α| ≤ 1 for every eigenvalue α of df^r_p under assumptions (1) bounded C_f and (2) Ker(κ_{n,p}) ⊂ Ker(gr_n f^{r*'}) for infinitely many n. It does so by: (i) using the filtration of distributions D_n(X)_p and Lemma 2.2/Corollary 2.3 to identify the top-graded action gr f^{*'} with the symmetric power of df_p, giving eigenvalues α^n on gr_n; (ii) pushing these to V′ via κ_{n,p}; and (iii) bounding the norms of the induced operators to obtain |α| ≤ ||C_f′||^{r/n} and hence |α| ≤ 1 as n ranges over infinitely many values. The candidate solution follows the same strategy: reduces to a fixed point; uses the intertwining ι′∘f^{*′} = C_f′∘ι′; identifies α^n on the graded piece via a Faà di Bruno/top-order argument; ensures the eigenvalues survive to the quotient via assumption (2); and finally bounds eigenvalues to reach a contradiction if |α|>1. Minor differences are cosmetic (the paper phrases the top-order action via the symmetric algebra map and uses a precise norm bound on gr_n C_f^{r′}, while the model bounds eigenvalues of C_f′ directly).