Classical dynamical systems

[Daniel]

Could you explain how this is represented?

Would it be something like:

\[
f^t(x) = \int_0^x e^{-tu} \, d\mu(u)\\
\]

I fail to understand what these dynamics mean?

I understand the Manifold comment. So let's assume that the manifold is \([0,\infty)\), and that we are creating some kind of iteration on here. Then we are writing:

\[
f^t(x)\\
\]

How are we using an integral interpretation?

I'm sorry, but I'm curious and I don't have that book or access to any article relating to it. Context would do me well?

Is this an extension of the Borel measure for regular iteration? Where there's a measure \(\mu\) such that:

\[
\sqrt{2} \uparrow \uparrow t = \int_0^\infty e^{-tx}\,d\mu(x)\\
\]

I'm sorry Daniel, but could you please elaborate.

First let me say I'm pushing my minimal understanding of physics to the limit here. Kousnetsov could probably explain it better.

Consider the iterated function \(f^t(x)\) in quantum mechanics. Let \(f(x) = Hx\) where x is an infinite column vector and H is an infinite matrix. The vector x is the initial state of the system and H is the laws of physics. Then the state of the system from instant to instant is \(x, H x, H^2 x, \cdots, H^tx.\)

The constraint that f is measure preserving translates to a system having a unit multiplier, that it is on the higher dimensional analog of the Shell-Thron boundary.

Second example is the Feynman Path Integral, the heart of quantum field theory. Once again the system is computed from one instant to the next. But here the function f integrates and then exponentiates the results. Without the integral the expression would reduce to tetration.