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This is more a comment than an answer but it's too long for a comment.

In ergodic theory (as opposed to dynamical systems), systems in which the acting group is not the integers or the reals have been widely studied, both for their own intrinsic interest and because of deep and striking applications to number theory and other areas. For example, the proof by Einsiedler, Katok, Lindenstrauss that the set of exceptions to Littlewood's conjecture has zero Hausdorff dimension uses an ergodic-theoretic result on higher rank group actions.

As soon as the acting group has higher rank (even in the simplest case, i.e. $\mathbb{Z}^2$), the study of their ergodic theoretic properties becomes dramatically more complicated, even if the phase space is as simple as possible (the circle). A famous example is Furstenberg's $\times 2\times 3$ problem: what are the measures which are simultaneously invariant under $x\to 2x \bmod 1$ and $x\to 3x\bmod 1$? (this corresponds to the action of $\mathbb{N}^2$ on $[0,1]$ given by $(a,b)\cdot x=2^a 3^b x\bmod 1$). It is suspected that there are very few invariant measures, which illustrates a general (conjectured or proved) phenomenon: higher rank dynamical systems tend to have few invariant measures, all of them with some algebraic structure. Indeed, Einsiedler, Katok and Lindenstrauss use such a rigidity result.

The book by Einsiedler and Ward "Ergodic thory (with a view towards number theory)" is an excellent reference for this general topic.

About your specific question, I'm not entirely convinced that analytic dependence on the time parameter is the right thing to look at since a dynamical system is a group action, i.e. only the additive structure of the complex numbers matters as far as the definition of dynamical system is concerned.

2 added 73 characters in body

This is more a comment than an answer but it's too long for a comment.

In ergodic theory (as opposed to dynamical systems), systems in which the acting group is not the integers or the reals have been widely studied, both for their own intrinsic interest and because of deep and striking applications to number theory and other areas. For example, the proof by Einsiedler, Katok, Lindenstrauss that the set of exceptions to Littlewood's conjecture has zero Hausdorff dimension uses an ergodic-theoretic result on higher rank group actions.

As soon as the acting group has higher rank (even in the simplest case, i.e. $\mathbb{Z}^2$), the study of their ergodic theoretic properties becomes dramatically more complicated, even if the phase space is as simple as possible (the circle). A famous example is Furstenberg's $\times 2\times 3$ problem: what are the measures which are simultaneously invariant under $x\to 2x \bmod 1$ and $x\to 3x\bmod 1$? (this corresponds to the action of $\mathbb{N}^2$ on $[0,1]$ given by $(a,b)\cdot x=2^a 3^b x\bmod 1$). It is suspected that there are very few invariant measures, which illustrates a general (conjectured or proved) phenomenon: higher rank dynamical systems tend to have few invariant measures, all of them with some algebraic structure. Indeed, Einsiedler, Katok and Lindenstrauss use such a rigidity result.

The book by Eisiedler Einsiedler and Ward "Ergodic thory (with a view towards number theory)" is an excellent reference for this general topic.

About your specific question, I'm not entirely convinced that analytic dependence on the time parameter is the right thing to look at since a dynamical system is a group action, i.e. only the additive structure of the complex numbers matters as far as the definition of dynamical system is concerned.

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This is more a comment than an answer but it's too long for a comment.

In ergodic theory (as opposed to dynamical systems), systems in which the acting group is not the integers or the reals have been widely studied, both for their own intrinsic interest and because of deep and striking applications to number theory and other areas. For example, the proof by Einsiedler, Katok, Lindenstrauss that the set of exceptions to Littlewood's conjecture has zero Hausdorff dimension uses an ergodic-theoretic result on higher rank group actions.

As soon as the acting group has higher rank (even in the simplest case, i.e. $\mathbb{Z}^2$), the study of their ergodic theoretic properties becomes dramatically more complicated, even if the phase space is as simple as possible (the circle). A famous example is Furstenberg's $\times 2\times 3$ problem: what are the measures which are simultaneously invariant under $x\to 2x \bmod 1$ and $x\to 3x\bmod 1$? (this corresponds to the action of $\mathbb{N}^2$ on $[0,1]$ given by $(a,b)\cdot x=2^a 3^b x\bmod 1$). It is suspected that there are very few invariant measures, which illustrates a general (conjectured or proved) phenomenon: higher rank dynamical systems tend to have few invariant measures, all of them with some algebraic structure.

The book by Eisiedler and Ward "Ergodic thory (with a view towards number theory)" is an excellent reference for this general topic.

About your specific question, I'm not entirely convinced that analytic dependence on the time parameter is the right thing to look at since a dynamical system is a group action, i.e. only the additive structure of the complex numbers matters as far as the definition of dynamical system is concerned.