Dynamical systems with multidimensional, complex and other exotic kinds of time spaces

Question

As one may know, a dynamical system can be defined with a monoid or a group action on a set, usually a manifold or similar kind of space with extra structure, which is called the phase space or state space of the dynamical system. The monoid or group doing the acting is what I call the time space of the dynamical system, and is usually the naturals, integers, or reals. Often, one may require the evolution map to be continuous, differentiable, etc.

But has anyone studied a generalization in which we allow the time space to be something more general & exotic, a multidimensional space like $\mathbb{R}^n$, $\mathbb{C}$ (viewed as "2-dimensional" by considering it to be like $\mathbb{R}^2$), etc.? I'm especially curious about the case where the time space is $\mathbb{C}$ and the phase space is $\mathbb{C}^n$ or another complex manifold and the map is required to be holomorphic in both its arguments, as that holomorphism provides a natural linkage between the two dimensions that lets us think of the complex time as a single 2-dimensional time as opposed to two real times (any dynamical system with a timespace of $\mathbb{R}^n$ can be decomposed into a bunch of mutually-commutative evolution maps with timespaces of $\mathbb{R}$). My questions are:

1. Is it true that the only dynamical system with phase-space $\mathbb{C}$ and time-space $\mathbb{C}$ where the evolution map is required to be holomorphic in both its arguments (the time and point to evolve) is the linear one given by $$\phi^t(z) = e^{ut} z + K \frac{1 - e^{ut}}{1 - e^u}, K, u \in \mathbb{C}$$ ? I suspect so, because an injective entire function is linear (in the sense of a “linear equation” not necessarily a “linear map”), and $\phi^{-t}$ must be the inverse of $\phi^t$. Thus, $\phi^t$ must have the form $a(t) z + b(t)$ with $a: \mathbb{C} \rightarrow \mathbb{C}$, $b: \mathbb{C} \rightarrow \mathbb{C}$. Am I right?

2. Are there any interesting (e.g. with complicated, even chaotic behavior) dynamical systems of this kind on $\mathbb{C}^n$? On more sophisticated complex manifolds? (for the phase space, that is) If so, can you provide an example? Or does the holomorphism requirement essentially rule this out? EDIT: I provide one below.

3. There is something else here, an interesting observation I made. Consider the above complex-time, holomorphic dynamical system. We can investigate the two prime behaviors represented by the real and imaginary times. We'll just set u = 1, K = 0 for here.

In “real time”, the dynamics looks like an “explosion” in the plane: all points “blast” away from z = 0 at exponentially increasing velocity.

In “imaginary time”, the result is cyclic motion, swirling around $z = 0$ with constant angular velocity that depends only on the distance of the point from $z = 0$.

But if we trace the contours of these two evolutions, formed from different points on the plane, and then superimpose them, we have contours intersecting in what looks like contours from a contour graph of the image of the complex plane under the function $\exp(z)$! Conversely, we could say it looks like a countour plot of $\log(z)$ with a cut along a ray from $0$. So, somehow “naturally” related to the dynamical system $\phi^t(z) = \exp(t) z$ is the function $\exp(z)$ (or, perhaps, $\log(z)$).

Note that my plotting facilities are unfortunately pretty limited, so I can't give really nice graphs with lots of contours, just a few selected ones taken from evolutions of various points in both real and imag times.

But we have another case. To see this, we must turn our attention away from a phase space given by the complex plane to one given by the Riemann sphere, $\hat{\mathbb{C}}$. In this case, we still have the dynamical system as above, but we have an additional class of dynamical systems given by the “Moebius transformations”, which include the above linear-function dynamical systems as a special case. One example is $$\phi^t(z) = \frac{(1 + e^{i\pi t}) z + (1 – e^{i\pi t})}{(1 – e^{i\pi t})z + (1 + e^{i\pi t})}$$. It is easy to check that this is indeed a Moebius transformation of the Riemann sphere. This map is holomorphic everywhere on the Riemann sphere. Note that for integer step t, the unit-step map is the reciprocal map.

Now we consider the contour lines of the real and imaginary evolution, as before. They look like this:

(Physics buffs may notice that the real evolution (concentric circles) reminds one of the lines of a magnetic field of a magnetic dipole (like a bar magnet), while the imaginary evolution (arcs joining at points) looks like the lines of an electric field of an electric dipole.)

Again, notice how the lines meet at right angles. It looks again like the image of the plane (or the Riemann sphere, perhaps?) under some function which may be holomorphic, though I'm not sure what that function is in this case. Is this the case? Is there such a function, with a special relation to this CTHDS in the same way as $\exp$ (or $\log$) is to the other?

But in any case, it appears that for a complex-time holomorphic dynamical system, or CTHDS, there exists an associated natural function. What is the significance of this function/map? How does it relate to the CTHDS? If you give me a CTHDS that I don't have a closed form for, can I find its natural map?

Answer 1

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.

Answer 2

I will address the case of $C$-action (as time). In the case when the phase space is also $C$, you are right, there are only Moebius transformations, which can be loxodromic (elliptic, hyperbolic) or parabolic. That is up to conjugacy by an arbitrary Moebius transformation you have $z\mapsto e^{\alpha t}z$ and $z\mapsto z+t$, where $t$ is your complex "time".

In the case when the phase space is $C^n$, we have a lot of possibilities. There is a hudge (infinite-dimensional) continuous group of analytic bijective transformations of $C^n$ onto itself. You can take any complex 1-parametric subgroup of this group. For $n=2$, there is a very nice paper of Milnor on this group.

Edit: Friedland, Shmuel; Milnor, John Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dynam. Systems 9 (1989), no. 1, 67–99.

Answer 3

I have seen talks about this, though somewhat over my head. But, yes absolutely, people do research such things. A. Katok is one name that comes to mind here. See this paper: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.2082

They consider actions of $\mathbb{Z}^k$ and $\mathbb{R}^k$ that generalize the usual time-like case of $\mathbb{Z}$ and $\mathbb{R}$.

B. Kalinin and V. Sadovskaya also work in this area, seeing their talks was how I was exposed to this field: http://www.southalabama.edu/mathstat/personal_pages/sadovska/Research/tns.pdf

A lot of the work I've been exposed to is about classifying such systems up to a conjugacy, and in particular they consider Anosov systems and see if they can find conjugacies in the general case to canonical examples.

This is not my expertise, though, so I can't say what this says about the $\mathbb{C}$ case you're interested in. But at least I can say, yes, indeed there is work in the higher-dimensional case.

Answer 4

This question is from long ago, but I will give it a try. I want to note that multi-dimensional times do occur "naturally", in some reasonable manner. As an example, take coupled map lattices, with two times involved: the actual time and the space translation, along which different systems may interact. In this setting (which is thus a $\mathbb Z^2$ action) it is easier to produce many invariant measures, say for short-ranged actions. In fact one can even develop a thermodynamic formalism, equilibrium and Gibbs measures, etc.

In the context of higher rank actions with some hyperbolicity there has also been some progress, notably with work of Kalinin and Katok where the existence of an invariant geometric structure is obtained for the first time from homotopy data.