2
$\begingroup$

Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. I do not assume global integrability in the sense of Liouville-Arnold - just that the phase space is foliated by invariant curves of the type $I(x)=c$, where $c$ is constant.

Must this map be necessarily area-preserving?

I do not know either how to prove this, or come up with a counter example. Similarly, for maps defined for $\mathbb{R}^{n}$...

$\endgroup$
  • 1
    $\begingroup$ Comment to the question (v1): The condition $I(x)=I(f(x))$ is trivially satisfied for any $f$ if $I$ is a constant function. $\endgroup$ – Qmechanic Oct 31 '15 at 0:59
  • $\begingroup$ Of course, but my question concerns the existence of a nontrivial integral @Qmechanic $\endgroup$ – Alex Oct 31 '15 at 17:00
  • $\begingroup$ Here's another counterexample. Let me identify the plane $\mathbb{R}^{2}\cong \mathbb{C}$. Assume $f:\mathbb{C}\to \mathbb{C}$ is of the form $f(z)=f_0(z)+i{\rm Im}(z)$, where $f_0:\mathbb{C}\to \mathbb{R}$ is a real-valued function chosen such that $f$ is not area-preserving, e.g. $f_0(z)=2{\rm Re}(z)$. Let $I:\mathbb{C}\to \mathbb{R}$ be the imaginary part $I(z):={\rm Im}(z)=I(f(z))$. $\endgroup$ – Qmechanic Oct 31 '15 at 19:00
  • $\begingroup$ It looks to me that your example can just be easily given in $\mathbb{R}^{2}$, without identification with $\mathbb{C}$, if we put $x_{1}=2x_{0},\quad y_{1}=y_{0}$, right? But then the variables are decoupled and we essentially have two 1-dimensional maps with trivial dynamics... @Qmechanic $\endgroup$ – Alex Nov 2 '15 at 0:37
  • $\begingroup$ $\uparrow$ Yes. $\endgroup$ – Qmechanic Nov 2 '15 at 0:40
0
$\begingroup$

No.

How about, e.g., the Hénon map with $a=1.25$ and $b=0.3$? It has an attractor, so it's not area-preserving, and this attractor is periodic, so the system is integrable.

Unless our definitions don't match, any integral map in $\mathbb{R}^n$ that has an attractor is a counter-example to integrability and area-preservation being linked.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.