2019/11/27 by Matthew D. Kvalheim, Kvalheim, Matthew D., Shai Revzen +1 · 2 citations
Physics and Astronomy · Biochemistry, Genetics and Molecular Biology · #Quantum chaos and dynamical systems #Protein Structure and Dynamics #Hemoglobin structure and function
paper · pdf · doi:10.48550/arxiv.1911.11996
We consider C1 dynamical systems having an attracting hyperbolic fixed\npoint or periodic orbit and prove existence and uniqueness results for Ck\n(actually Ck,\α\loc) linearizing semiconjugacies -- of which\nKoopman eigenfunctions are a special case -- defined on the entire basin of\nattraction. Our main results both generalize and sharpen Sternberg's Ck\nlinearization theorem for hyperbolic sinks, and in particular our corollaries\ninclude uniqueness statements for Sternberg linearizations and Floquet normal\nforms. Using our main results we also prove new existence and uniqueness\nstatements for Ck Koopman eigenfunctions, including a complete\nclassification of C^\∞ eigenfunctions assuming a C^\∞ dynamical\nsystem with semisimple and nonresonant linearization. We give an intrinsic\ndefinition of "principal Koopman eigenfunctions" which generalizes the\ndefinition of Mohr and Mezi 'c for linear systems, and which includes the\nnotions of "isostables" and "isostable coordinates" appearing in work by\nErmentrout, Mauroy, Mezi 'c, Moehlis, Wilson, and others. Our main results\nyield existence and uniqueness theorems for the principal eigenfunctions and\nisostable coordinates and also show, e.g., that the (a priori non-unique)\n"pullback algebra" defined in citemohr2016koopman is unique under certain\nconditions. We also discuss the limit used to define the "faster" isostable\ncoordinates in citewilson2018greater,monga2019phase in light of our main\nresults.\n