From b515d36b798a8e37be5992db186fc5236eebc434 Mon Sep 17 00:00:00 2001 From: Sergiu Ivanov Date: Tue, 9 Apr 2024 09:00:29 +0200 Subject: [PATCH] Define the state graph. --- deal.tex | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/deal.tex b/deal.tex index ee82d1a..681fe50 100644 --- a/deal.tex +++ b/deal.tex @@ -534,7 +534,10 @@ $t$ steps, while $\Phi(s, -t)$ can be seen as the state the system to see the discrete-time dynamical systems as the function $F : X \to X$, giving the state to which the dynamical system transitions from state $s \in X$: $F(s) = \Phi(s, 1)$ -(e.g,~\cite[Chapter~1]{Riva22}). +(e.g,~\cite[Chapter~1]{Riva22}). $F$~generates the possibly infinite +state graph of the discrete-time dynamical system, whose nodes are the +states, and there is a directed edge from state $s_1$ to state $s_2$ +if $s_2 = F(s_1)$. Given a dynamical system $\Phi : X \times T \to X$ and a state $s \in X$, it is possible to construct the restricted function