From 8fcedc5c5e81c933175de900a56b31f3e393d585 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Wed, 2 Aug 2023 14:06:56 +0300 Subject: [PATCH] updated bad x in dynamical systems --- Mathematics/4th/Dynamical_systems/Dynamical_systems.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex b/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex index 78a5ca8..f2a3036 100644 --- a/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex +++ b/Mathematics/4th/Dynamical_systems/Dynamical_systems.tex @@ -920,7 +920,7 @@ Let $f : I\rightarrow I$ be a function. The iteration $x_{n+1}=f(x_n)$ is \emph{topologically transitive} if for any pair of open subsets $U,V\subseteq I$, $\exists k\in\NN$ such that $f^k(U)\cap V\ne\varnothing$. \end{definition} \begin{definition} - Let $f : I\rightarrow I$ be a function. The iteration $x_{n+1}=f(x_n)$ has \emph{sensitive dependence on initial conditions} on $I$ if $\exists\delta >0$ such that for each $x\in I$ and any neighborhood $N$ of x, exists $y \in N$ and $n \geq 0$ such that $\abs{f^n(x)-f^n(y)} > \delta$. + Let $f : I\rightarrow I$ be a function. The iteration $x_{n+1}=f(x_n)$ has \emph{sensitive dependence on initial conditions} on $I$ if $\exists\delta >0$ such that for each $x\in I$ and any neighborhood $N$ of $x$, exists $y \in N$ and $n \geq 0$ such that $\abs{f^n(x)-f^n(y)} > \delta$. \end{definition} \begin{lemma} Let $f : I\rightarrow I$ be a function such that the iteration $x_{n+1}=f(x_n)$ is topologically transitive. Then, it has a dense orbit.