As usual in those matters, this is more a long comment than a full comprehensive answer.<br>
I am not aware of works applying directly the theory of dynamical systems to the analytic theory of numbers. Nevertheless I know that the Siegel-Shidlovskii theory of transcendental numbers relies on the theory of homogeneous linear ordinary differential equations of polynomial, or more generally rational, coefficients which is undoubtedly a chapter of this theory.<br> It seems that the roots of this theory lie in an ancient observation by [A. M. Legendre](https://en.wikipedia.org/wiki/Adrien-Marie_Legendre) (according to [1], chapter 1, §4, p. 7 and  [2], §4, p. 39): by considering the power series 
$$
f_\alpha(x)=\sum_{n=0}^\infty \frac{x^n}{n! \alpha(\alpha+1)\cdots(\alpha+n-1)},\quad \alpha\neq 0, -1, -2, \ldots,
$$
which is an entire solution of the following linear ODE with polynomial coefficient
$$
x f^{\prime\prime}(x)+\alpha f^{\prime}(x)= f(x),
$$
Legendre noted that *the ratio $f_\alpha(x)/ f^{\prime}_\alpha(x)$ is irrational for any rational $x\neq 0$ and $\alpha$* satisfying the above requirements. Later [Erik Stridsberg](https://www.tandfonline.com/doi/abs/10.1080/03461238.1951.10432128) proved the irrationality of the dividend $f_\alpha(x)$ and of the divisor $f^{\prime}_\alpha(x)$ for the same values of $x$ and $\alpha$, but  decisive step in this approach was done by [Carl Ludwig Siegel](https://en.wikipedia.org/wiki/Carl_Ludwig_Siegel) in 1949.<br>
Siegel defined the class of $E$-functions by using the following linear system of ODEs with rational coefficients
$$
\frac{\mathrm{d}}{\mathrm{d} x}
\begin{pmatrix} 
f_1(x)\\
\vdots\\
f_k(x)\\
\vdots\\
f_m(x)
\end{pmatrix} = 
\begin{pmatrix}
Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \\
Q_{2,1}(x) & Q_{2,2}(x) & \ldots &Q_{2,m}(x) \\
\vdots & \vdots & \ddots & \vdots \\
Q_{m,1}(x) & Q_{m,2}(x) & \ldots &Q_{m,m}(x)
\end{pmatrix}
\begin{pmatrix} 
f_1(x)\\
\vdots\\
f_k(x)\\
\vdots\\
f_m(x)
\end{pmatrix},
$$
and was able to prove, assuming suitable hypotheses on the coefficients and on its solutions $\big(f_1(x), \ldots, f_m(x)\big)$ are verified, that all the $m$ numbers $f_1(\alpha), \ldots, f_m(\alpha)$ where $\alpha$ is and algebraic number which is not $0$ nor it is a pole of the functions $\{Q_{i,j}\}_{1\le i,j\le m}$ are algebraically independent. Later on [Andrei Borisovich Shidlovskii](http://www.numbertheory.org/ntw/obituaries/RMS/shidlovsky.pdf), was able to simplify in a meaningful way the hypothesis under which the results of Siegel holds, and thus develop further Siegel's original approach. The two references cited below should be more than sufficient to give a sketch of the theory.

**References**

[1] Andrei Borisovich Shidlovskii, *Transcendental numbers*. With a foreword by W. Dale Brownawell. Translated from the Russian by Neal Koblitz. (English) De Gruyter Studies in Mathematics, 12. Berlin-New York: Walter de Gruyter, pp. xix+466 (1989), ISBN:3-11-011568-9, [MR1033015](https://mathscinet.ams.org/mathscinet-getitem?mr=1033015), [Zbl 0689.10043](https://zbmath.org/0689.10043).

[2] Naum Il’ich Fel’dman and Andrei Borisovich Shidlovskii, "[The development and present state of the theory of transcendental numbers](https://www.mathnet.ru/eng/rm5754)" (English. Russian original) Russian Mathematical Surveys 22, No. 3, 1-79 (1967); translation from Uspekhi Matematicheskikh Nauk [N. S.] 22, No. 3(135), 3-81 (1967), [MR214551](https://mathscinet.ams.org/mathscinet-getitem?mr=214551), [Zbl 0178.04801](https://zbmath.org/0178.04801).