What does it mean by "dynamics of a sequence" mathematically?

Question

Consider the following paragraph from the topic named sequential models from the textbook titled Dive into Deep Learning

Both cases raise the obvious question of how to generate training data. One typically uses historical observations to predict the next observation given the ones up to right now. Obviously we do not expect time to stand still. However, a common assumption is that while the specific values of might change, at least the dynamics of the sequence itself will not. This is reasonable, since novel dynamics are just that, novel and thus not predictable using data that we have so far. Statisticians call dynamics that do not change stationary.

Here sequence refers to $x_1, x_2, x_3, \cdots, x_t$. Say the stock price of a company.

What does it mean rigorously by the dynamics of a sequence in statistics?

Answer 1

I don't think there is any rigorous mathematical definition of dynamic, or to phrase it better, the definition is embedded in the definition of dynamical system itself (which would be probably a better term to use in place of the generic "sequence"):

A dynamical system is formally defined as a state space $X$, a set of time points $T$, and a rule $R$ that specifies how the state evolves with time. The rule $R$ is a function whose domain is $X×T$ and whose codomain is $X$, i.e., $R:X×T→X$. The rule function $R$ means that the $R$ takes two inputs, $R=R(x,t)$, where $x∈X$ is an initial state and $t∈T$ is a specific time step.

I think that what the paragraph states is that we assume the rule funcion R to remain the same, cause that's what we're trying to learn.

As a final note, there are out there also classifications of different non-linear dynamical systems, based on the property of $R$.