Simple proofs and extensions of a result of L. D. Pustylnikov on the nonautonomous Siegel theorem

摘要

AbstractWe present simple proofs of a result of L.D. Pustylnikov extending to nonautonomous dynamics the Siegel theorem of linearization of analytic mappings. We show that if a sequencefnof analytic mappings ofCdhas a common fixed pointfn(0) = 0, and the mapsfnconverge to a linear mappingA∞ so fast that$$sumlimits_n {{{left| {{f_m} - {A_infty }} ight|}_{Linfty left( B ight)}} < infty } $$∑n‖fm?A∞‖L∞(B)/mo><mi>∞</mi>$${A_infty } = diagleft( {{e^{2pi i{omega _1}}},...,{e^{2pi i{omega _d}}}} ight)omega = left( {{omega _1},...,{omega _q}} ight) in {mathbb{R}^d},$$A∞=diag(e2πiω1,...,e2πiωd)ω=(ω1,...,ωq)∈?d,thenfnis nonautonomously conjugate to the linearization. That is, there exists a sequencehnof analytic mappings fixing the origin satisfying$${h_{n + 1}} circ {f_n} = {A_infty }{h_n}.$$hn+1?fn=A∞hn.The key point of the result is that the functions hn are defined in a large domain and they are bounded. We show that$${sumolimits_n {left| {{h_n} - Id} ight|} _{Linfty (B)}} < infty .$$