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2013
Impact Factor

Rafael de la Llave

686 Cherry St., Atlanta GA 30332-0160, USA
Georgia Institute of Technology, School of Mathematics

Publications:

de la Llave R.
Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions
2018, vol. 23, no. 1, pp.  1-11
Abstract
A well-known result in complex dynamics shows that if the iterates of an analytic map are uniformly bounded in a complex domain, then the map is analytically conjugate to a linear map. We present a simple proof of this result in any dimension. We also present several generalizations and relations to other results in the literature.
Keywords: analytic maps, linearization
Citation: de la Llave R.,  Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 1-11
DOI:10.1134/S156035471801001X
de la Llave R.
Simple Proofs and Extensions of a Result of L. D. Pustylnikov on the Nonautonomous Siegel Theorem
2017, vol. 22, no. 6, pp.  650–676
Abstract
We 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 sequence $f_n$ of analytic mappings of ${\mathbb C}^d$ has a common fixed point $f_n(0) = 0$, and the maps $f_n$ converge to a linear mapping $A_\infty$ so fast that $$ \sum_n \|f_m - A_\infty\|_{\mathbf{L}^\infty(B)} < \infty $$ $$ A_\infty = \mathop{\rm diag}( e^{2 \pi i \omega_1}, \ldots, e^{2 \pi i \omega_d}) \qquad \omega = (\omega_1, \ldots, \omega_q) \in {\mathbb R}^d, $$ then $f_n$ is nonautonomously conjugate to the linearization. That is, there exists a sequence $h_n$ of analytic mappings fixing the origin satisfying \[ h_{n+1} \circ f_n = A_\infty h_{n}. \] The key point of the result is that the functions $h_n$ are defined in a large domain and they are bounded. We show that $\sum_n \|h_n - \mathop{\rm Id} \|_{\mathbf{L}^\infty(B)} < \infty$.
We also provide results when $f_n$ converges to a nonlinearizable mapping $f_\infty$ or to a nonelliptic linear mapping.
In the case that the mappings $f_n$ preserve a geometric structure (e.g., symplectic, volume, contact, Poisson, etc.), we show that the $h_n$ can be chosen so that they preserve the same geometric structure as the $f_n$.
We present five elementary proofs based on different methods and compare them. Notably, we consider the results in the light of scattering theory. We hope that including different methods can serve as an introduction to methods to study conjugacy equations.
Keywords: nonautonomous linearization, scattering theory, implicit function theorem, deformations
Citation: de la Llave R.,  Simple Proofs and Extensions of a Result of L. D. Pustylnikov on the Nonautonomous Siegel Theorem, Regular and Chaotic Dynamics, 2017, vol. 22, no. 6, pp. 650–676
DOI:10.1134/S1560354717060053

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