updated bad x in dynamical systems

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.