final updated advanced dyn syst,

Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex

@@ -136,7 +136,7 @@
     $$
       \lim_{n\to\infty}\int_{\TT^2} \exp{2\pi i(\transpose{(\vf{\tilde{A}}^n)}\vf{p}+\vf{q})\cdot \vf{x}}\dd{\vf{x}}=0
     $$
-    So for any $\vf{p}, \vf{q}\in\ZZ^2$ we have the equality. Then, we use that any function nice enough can be approximated with its Fourier series.
+    So for any $\vf{p}, \vf{q}\in\ZZ^2$ we have the equality. Then, we use that any function nice enough can be approximated uniformly with the Féjer means of the Fourier series (see \mnameref{MA:fejerthm}).
   \end{proof}
   \begin{theorem}
     On the torus $\TT^2$ there exist two direction fields invariant with respect to the automorphism $\vf{\tilde{A}}$. The integral curves of each of these directions fields are everywhere dense on the torus. The automorphism $\vf{\tilde{A}}$ converts the integral curves of each field into integral curves of the same field, expanding by $\lambda_+$ for the first field and contracting by $\lambda_-$ for the second.
@@ -150,17 +150,17 @@
     be the expanding and contracting curves and let $\vf{\xi}_{\vf{x}}=\im(\vf\gamma_+)$, $\vf{\eta}_{\vf{x}}=\im(\vf\gamma_-)$ be the corresponding direction fields. The density of the curves is a consequence of the density of orbits in rotation maps in the circle with irrational angle.
   \end{proof}
   \begin{definition}
-    Let $\vf{A},\vf{B}:\TT^2\rightarrow \TT^2$ be $\mathcal{C}^1$ functions. We say that $B$ is \emph{$\mathcal{C}^0$-close} to $\vf{A}$ if for all $\varepsilon>0$:
+    Let $\vf{A},\vf{B}:\TT^2\rightarrow \TT^2$ be $\mathcal{C}^1$ functions and $\varepsilon>0$. We say that $B$ is \emph{$\mathcal{C}^0$-$\varepsilon$-close} to $\vf{A}$ if:
     $$
-      \sup_{\vf{x}\in \TT^2}\norm{\vf{A}(\vf{x})-\vf{B}(\vf{x})}<\varepsilon
+      \sup_{\vf{x}\in \TT^2}\norm{\vf{B}(\vf{x})-\vf{A}(\vf{x})}<\varepsilon
     $$
-    We say that $\vf{B}$ is \emph{$\mathcal{C}^1$-close} to $\vf{A}$ if $\vf{B}$ is $\mathcal{C}^0$-close to $\vf{A}$ and for all $\varepsilon>0$:
+    We say that $\vf{B}$ is \emph{$\mathcal{C}^1$-$\varepsilon$-close} to $\vf{A}$ if they are $\mathcal{C}^0$-$\varepsilon$-close and:
     $$
-      \sup_{\vf{x}\in \TT^2}\norm{\vf{D}\vf{A}(\vf{x})-\vf{D}\vf{B}(\vf{x})}<\varepsilon
+      \sup_{\vf{x}\in \TT^2}\norm{\vf{DB}(\vf{x})-\vf{DA}(\vf{x})}<\varepsilon
     $$
   \end{definition}
   \begin{theorem}[Structal stability]
-    Let $\vf{B}$ be a diffeomorphism on $\TT^2$ which is $\mathcal{C}^1$-close to $\vf{\tilde{A}}$. Then, $\vf{B}$ is $\mathcal{C}^0$-conjugate to $\vf{\tilde{A}}$.
+    Let $\vf{B}$ be a diffeomorphism on $\TT^2$ which is $\mathcal{C}^1$-$\varepsilon$-close to $\vf{\tilde{A}}$. Then, $\vf{B}$ is $\mathcal{C}^0$-conjugate to $\vf{\tilde{A}}$.
   \end{theorem}
   \begin{proof}
     We need to find a $\mathcal{C}^0$-conjugacy $\vf{H}$ between $\vf{B}$ and $\vf{\tilde{A}}$. Since, $\vf{B}$ is $\mathcal{C}^1$-close to $\vf{\tilde{A}}$, we may expect that both $\vf{H}$ and $\vf{B}$ are small perturbations of the identity and $\vf{\tilde{A}}$ respectively. So set $\vf{H}=\vf{I}+\vf{h}$ and $\vf{B}=\vf{\tilde{A}}+\vf{b}$. Then, we want to find $\vf{h}$ and $\vf{b}$ such that:
@@ -427,7 +427,7 @@
     Taking limits, we have that $\rho(f)=\rho(g)$, as $\varphi$ is bounded.
   \end{proof}
   \begin{remark}
-    Note that the proof also works even if $\exists h=\id+\varphi\in \mathcal{C}(\TT^1)$ such that $h\circ f=g\circ h$ (i.e.\ $H$ is only a semi-conjugacy).
+    Note that the proof also works even if $\exists h=\id+\varphi\in \mathcal{C}(\TT^1)$ such that $h\circ f=g\circ h$ (i.e.\ $H$ is only a \textit{special} semi-conjugacy).
   \end{remark}
   \subsubsection{Rotation number and invariant measure}
   \begin{definition}