Why study probabilities in deterministic dynamical systems?

Question

When studying hyperbolic dynamics and ergodic theory, one often come with probabilities (measures that give to the total space measure 1) even in deterministic systems. Do these have some intuitive interpretation as a probability? Because it seems rather strange to me to put probabilities in deterministic systems, since it hardly seems to mean anything: any probability that you would put on it seems to be arbitrary and not actually meaning anything about the system. Or should it just be viewed (intuitively) as a usual measure (that is finite so you can make calculations easier but normalizing it)?

Answer 1

This is a sketch i.e. WIP. I will use smooth space, which may bring some intuition to the topic.

At first, think the ergodic variable $z$ as the input vector on a SMOOTH map $f(x, z)$. The Taylor expansion of flow trajectory $\phi(t, x, t_0, x_0)$ has the homogeneous term respective to $f(x, 0)$, the so-called exponential map, and the contribution respective to ergodic components $f_{z_i}$, for index $i \in {\mathbb{N}^*_{\leq n_z}}$.

$$\mathcal{A}=\{u \in \mathcal{U} \, \lvert \, f_z = f(\cdot, z)\}$$

Let us call $G_j$ as the set of vector fields such that $G_0$ is given by set $\mathcal{A}$. There is a stop condition for this iteration, which is given by the the minimal set spanned by vector fields of $\{G_0, \cdots, G_m\}$ that spans the greatest vector subspace of tangent vector space $T_x\mathcal{X}$, we call it the accessible set $\mathcal{R} \subset \mathcal{X}$. There are other formal intuitive concepts like accessibility in time $t$ from point $p \in \mathcal{X}$

$$\{\omega_k \in G_{j-1} \, \lvert \, [\omega_k, f_{z_i}], \forall f_{z_i} \in \mathcal{A} \}$$

Therefore, there is an expected value for trajectory

$$\phi(t, x, t_0, x_0) \approx \int_{z \in \mathcal{Z}} e^{(f_x + \sum\limits_{\theta \in \mathcal{R}} \theta) t} \,p(z) \, dz $$

I hope, I did not offend anybody with this squiggle. :-)

Answer 2

The introduction of measure-based methods into the study of deterministic dynamical systems is (in my subjective opinion) not necessarily to express a probabilistic interpretation of the dynamics, but to come up with a mathematically rigorous mechanism to describe the "proportion" of a subset of elements/regions in your dynamical system. For example, tracking the measure of the subset of elements that wind up in a specific region of space over time can be interpreted as quantifying "how much" of your system eventually winds up in that region.

This was (and is!) also done through topological formulations, but measure-based methods tend to fare far better in describing dynamics on continua, as they avoid certain fundamental flaws in the construction of the former (see for, example, this preprint I wrote on an example of these pathologies.)