A Simple Proof of Gevrey Estimates for Expansions of Quasi-Periodic Orbits: Dissipative Models and Lower-Dimensional Tori

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.