Rafael de la Llave
686 Cherry St., Atlanta GA 303320160, USA
Georgia Institute of Technology, School of Mathematics
Publications:
Bustamante A., de la Llave R.
A Simple Proof of Gevrey Estimates for Expansions of QuasiPeriodic Orbits: Dissipative Models and LowerDimensional Tori
2023, vol. 28, nos. 45, pp. 707730
Abstract
We consider standardlike/Froeschl\'e dissipative maps
with a dissipation and nonlinear perturbation. That is,
\[
T_\varepsilon(p,q) = \left(
(1  \gamma \varepsilon^3) p + \mu + \varepsilon V'(q),
q + (1  \gamma \varepsilon^3) p + \mu + \varepsilon V'(q) \bmod 2 \pi \right)
\]
where $p \in {\mathbb R}^D$, $q \in {\mathbb T}^D$ are the dynamical
variables. We fix a frequency $\omega \in {\mathbb R}^D$ and study the existence of
quasiperiodic orbits. When there is dissipation, having
a quasiperiodic orbit of frequency $\omega$ requires
selecting the parameter $\mu$, called \textit{the drift}.
We first study the Lindstedt series (formal power series in $\varepsilon$) for quasiperiodic orbits with $D$ independent frequencies and the drift when $\gamma \ne 0$.
We show that, when $\omega$ is
irrational, the series exist to all orders, and when $\omega$ is Diophantine,
we show that the formal Lindstedt series are Gevrey.
The Gevrey nature of the Lindstedt series above was shown
in~\cite{BustamanteL22} using a more general method, but the present proof is
rather elementary.
We also study the case when $D = 2$, but the quasiperiodic orbits
have only one independent frequency (lowerdimensional tori).
Both when $\gamma = 0$ and when $\gamma \ne 0$, we show
that, under some mild nondegeneracy conditions on $V$, there
are (at least two) formal Lindstedt series defined to all orders
and that they are Gevrey.

de la Llave R.
Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions
2018, vol. 23, no. 1, pp. 111
Abstract
A wellknown 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.

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. 