Rafael de la Llave

We consider standard-like/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 quasi-periodic orbits. When there is dissipation, having a quasi-periodic 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 quasi-periodic 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 quasi-periodic orbits have only one independent frequency (lower-dimensional 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.

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.

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.