As usual in those matters, this is more a long comment than a full comprehensive answer.
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.
It seems that the roots of this theory lie in an ancient observation by A. M. 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 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 in 1949.
Siegel defined the class of $E$-functions by using the following linear system of ODEs with rational coefficients $$ \frac{\mathrm{d}}{\mathrm{d} x} \begin{pmatrix} f_1(x)\\ \vdots\\ f_k(x)\\ \vdots\\ f_m(x) \end{pmatrix} = \begin{pmatrix} Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \\ Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \\ \vdots & \vdots & \ddots & \vdots \\ Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \end{pmatrix} \begin{pmatrix} f_1(x)\\ \vdots\\ f_k(x)\\ \vdots\\ f_m(x) \end{pmatrix}, $$ and was able to prove that, 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, 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, Zbl 0689.10043.

[2] Naum Il’ich Fel’dman and Andrei Borisovich Shidlovskii, "The development and present state of the theory of transcendental numbers" (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, Zbl 0178.04801.

As usual in those matters, this is more a long comment than a full comprehensive answer.
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.
It seems that the roots of this theory lie in an ancient observation by A. M. 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 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 in 1949.
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, 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, Zbl 0689.10043.

[2] Naum Il’ich Fel’dman and Andrei Borisovich Shidlovskii, "The development and present state of the theory of transcendental numbers" (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, Zbl 0178.04801.

As usual in those matters, this is more a long comment than a full comprehensive answer.
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.
It seems that the roots of this theory lie in an ancient observation by A. M. 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 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 in 1949.
Siegel defined the class of $E$-functions by using the following linear system of ODEs with rational coefficients $$ \frac{\mathrm{d}}{\mathrm{d} x} \begin{pmatrix} f_1(x)\\ \vdots\\ f_k(x)\\ \vdots\\ f_m(x) \end{pmatrix} = \begin{pmatrix} Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \\ Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \\ \vdots & \vdots & \ddots & \vdots \\ Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \end{pmatrix} \begin{pmatrix} f_1(x)\\ \vdots\\ f_k(x)\\ \vdots\\ f_m(x) \end{pmatrix}, $$ and was able to prove that, assuming suitable hypotheses on the coefficients and on its solutions $(f_1(x), \ldots, f_m(x)$ 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, 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, Zbl 0689.10043.

[2] Naum Il’ich Fel’dman and Andrei Borisovich Shidlovskii, "The development and present state of the theory of transcendental numbers" (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, Zbl 0178.04801.

As usual in those matters, this is more a long comment than a full comprehensive answer.
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.
It seems that the roots of this theory lie in an ancient observation by A. M. 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 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 in 1949.
Siegel defined the class of $E$-functions by using the following linear system of ODEs with rational coefficients $$ \frac{\mathrm{d}}{\mathrm{d} x} \begin{pmatrix} f_1(x)\\ \vdots\\ f_k(x)\\ \vdots\\ f_m(x) \end{pmatrix} = \begin{pmatrix} Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \\ Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \\ \vdots & \vdots & \ddots & \vdots \\ Q_{1,1}(x) & Q_{1,2}(x) & \ldots &Q_{1,m}(x) \end{pmatrix} \begin{pmatrix} f_1(x)\\ \vdots\\ f_k(x)\\ \vdots\\ f_m(x) \end{pmatrix}, $$ and was able to prove that, 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, 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, Zbl 0689.10043.

[2] Naum Il’ich Fel’dman and Andrei Borisovich Shidlovskii, "The development and present state of the theory of transcendental numbers" (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, Zbl 0178.04801.