Invariant curves induce invariant regions in discrete, 2D dynamical systems?

Question

Consider a discrete dynamical system $x_{k+1} = f(x_k)$, where $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, sufficiently smooth, and let $C \subseteq \mathbb{R}^2$ be an invariant, closed curve in the phase space.

By Jordan's theorem, $C$ gives rise to two more connected regions, the interior $C_{\text{int}}$ and the exterior $C_\text{ext}$ of $C$.

If the system were continuous, then $C_{\text{int}}$ and $C_\text{ext}$ would be invariant regions, as the phase spaces of a continuous system can be shown to be a disjoint union of the orbits of the system - which means, that in particular the orbits cannot cross $C$.

But if the system is, as stated, discrete, this argument does not apply anymore, as it would be a priori conceivable that between two discrete time steps a jump across $C$ occurs (though perhaps some really simple argument that I just fail to see prevents this). Is there any other way to show that in the discrete case $C_\text{int}$ and $C_\text{ext}$ are invariant? Or does perhaps a counterexample exist?

Answer 1

This is a counter-example over the punctured plane.

Counter-example. [$f$ smooth over $\mathbb{R}^2\setminus \left\{0\right\}$] Consider the family of maps or discrete-time dynamical systems (indexed by $\alpha>0$) $f_{\alpha}\,:\mathbb{R}^{2}\setminus \left\{0\right\}\longrightarrow \mathbb{R}^2$ given by $f_{\alpha}(x,y)=\left(\frac{x}{\|(x,y)\|^{\alpha}},\frac{y}{\|(x,y)\|^{\alpha}}\right)$, where $\|\cdot \|$ is the Euclidean norm. Remark that $\|f_{\alpha}(x,y)\|=\|(x,y)\|^{1-\alpha}$. The unit circle $S$ is invariant w.r.t. any map $f_{\alpha}$ in the family. If $\alpha>1$, the dynamical system $f_{\alpha}$ contracts: if $(x_k,y_k)$ lies outside the unit circle (i.e., $\|(x_k,y_k)\|>1$), then $\|f_{\alpha}(x_k,y_k)\|=\|(x_k,y_k)\|^{1-\alpha}<1$, i.e., $f_{\alpha}(x_k,y_k)$ lies inside the unit circle. In other words, $S_{{\sf ext}}$ is not invariant w.r.t. any $f_{\alpha}$ with $\alpha>1$. On the other hand, if $\alpha<1$, the dynamical system $f_{\alpha}$ expands and $S_{{\sf int}}$ is not invariant w.r.t. any $f_{\alpha}$ with $\alpha<1$.

Answer 2

You state that $f$ is sufficiently smooth. However, the essential topological property, which you do not address, is whether or not the function $f$ is injective. This makes all the difference.

The time $t$ flow of a vectorfield (without finite time blowup) is a diffeomorphism. In particular it is injective and this prevent flowlines from crossing, whence the phenomena you mention for that case. In the case of a continuous map $f$, if the image of the interior contains points in the interior as well as in the exterior then being continuous it also contains points on the separating closed curve $C$ so it cannot be injective (Jordan Curve Theorem). In this case you can say very little about invariant sets.

Instead, let us assume that $f$ is injective and continuous and maps the Jordan curve $C$ surjectively onto itself. Then suddenly you can say a lot about invariant domains. I claim that each of $C_{\rm int}$ and $C_{\rm ext}$ are both invariant domains and moreover, $C_{\rm int}$ must be completely invariant, i.e. $f(C_{\rm int})=C_{\rm int}$. Note that we only assumed injectivity and continuity of $f$.

Proof: Invariance of Domain implies that $f$ is an open map of the Euclidean plane, i.e. a homeomorphism onto its image $I=f({\Bbb R}^2)$. This implies that $f$ maps a simply connected domain in the plane onto a simply connected domain. $D=C\cup C_{\rm int}$ is simply connected. Thus, any closed curve $\gamma$ may be continuously deformed within $D$ into a point. Thus, $f(D)$ is also contractible and this is not possible if $f$ maps $C_{\rm int}$ into $C_{\rm ext}$ and preserves $C$ (since $C$ would not be contractible within the image). Thus $f(D)\subset D$. And since $f(D)$ is simply connected and contains $C$ it can not omit any points in $D$, so we must have $f(D)=D$. Then $f$ must also map the exterior into itself, although not necessarily surjectively.

The above is related to the topology of the plane. On e.g. the punctured plane or on the sphere you may have diffeos exchanging interior and exterior.