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I'm reading Vlademir A. Zorich's Mathmatical Analysis I, meeting exercise question as following:

  1. a) A set $S \subset X$ is stable with respect to a mapping $f:X \rightarrow X$ if $f(S) \subset S$. Describe the sets that are stable with respect to a shift of the plane by a given vector lying in the plane.

    b) A set $I \subset X$ is invariant with respect to a mapping $f:X \rightarrow X$ if $f(I)=I$. Describe the sets that are invariant with respect to rotation of the plane about a fixed point.

    c) A point $p \in X$ is a fixed point of a mapping $f:X \rightarrow X$ if $f(p)=p$. Verify that any composition of a shift, a rotation, and a similarity transformation of the plane has a fixed point, provided the coefficient of the similarity transformation is less than 1.

    d) Regarding the Galilean and Lorentz transformation as mappings of the plane into itself for which the point with coordinates $(x,t)$ maps to the point with coordinates $(x',t')$, find the invariant sets of those transformations.

For sub-question a): I try to describe the stable sets of shift transformation as union sets looking like $\cup A_i$, where $A_i=\{p_i, f(p_i), f^2(p_i), f^3(p_i), ..., f^n(p_i),...\}$, $p_i$ is (arbitrary) point in the plane $X$ and $f$ denotes the shift transformation. In other words, a stable set of the shift transformation is a set of series of points, each of them consists of an initial point $p_i$ and all the points can be made by shifting $p_i$. But I think it is tedious and a bit ambiguous, and I find it difficult to describe the invariant set of the shift by this way. Will there be a clearer description? And how can I distinguish the invariant set from the stable set of the shift transformation?

For sub-question b) and d): I think the answer to b) is simply any set centrosymmetric with respect to the fixed point and the rotation angle. But when I turning to sub-question d), soon I failed to point out the invariant sets for those two transformations. What should I start with to consider this sub-question? Will there be a way to find the invariant set for an arbitrary linear transformation in $\mathbb{R^2}$?

I have no trouble with sub-question c), through it has been asked in this website(Fixed point in plane transformation.). This makes me confused: am I actually get something totally wrong and those questions are quite trivial?

My English may be somehow strange. I hope this won't make my question unclear.

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  • $\begingroup$ Haven't you asked such a question before? It sounds familiar. Ah yes, here it is… I think it would have been better to edit and undelete that one, but never mind. $\endgroup$ – MvG Aug 31 '15 at 10:07
  • $\begingroup$ @MvG Thank you for your suggestion...and would you have some idea on this question? I'm still stucking yet... $\endgroup$ – Asydot Aug 31 '15 at 10:14
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I think you nicely understoof the minimal stable sets for a), i.e. the sets $A_i$ from which you can not remove anything if you want to keep it stable. It is possible to describe some unions of such sets nicely, but if you want to describe all stable sets, you have to go to that level you used.

What's the difference between stable and invariant? With a stable set you can have an application of $f$ remove points (i.e. $x\in S$ but $x\not\in f(S)$), while for an invariant set this cannot happen. Which means that an invariant set is also stable under $f^{-1}$, at least if such an operation is well-defined as it is here. That's because $x\in f(S)$ is equivalent to $f^{-1}(x)\in S$ in that case. Combine that with your previous intuition and you have the invariant sets.

I think b) is a bit unclear: are you to describe sets invariant under a single given rotation about that fixed point, or invariant under all the possible rotations around it? The formalism you have (with just a single function $f$) suggests the former. Which makes the situation very similar to a). You might want to distinguish cases where the angle of rotation is a rational multipple of $\pi$, so you eventually end up where you started, from those where this is not the case. The latter would require quite a bit of number theory, I think, at least if you want to again describe minimal sets, as opposed to some unions of these which can be described nicely.

I agree that c) is the easiest of these questions, so it's no surprise that you solved that easier than the others. Computing fixed points will in general be easier than computing stable or fixed sets, simply because the former have just a single point as answer, while the latter could be point sets of pretty much arbitrary shape. The definition with the $A_i$ you gave at the beginning of your question is about as general as you can get here, I think. Unless you know more about the transformation in question.

I have a hunch about what kinds of invariant sets they'd expect for d). If so, then it's not the minimal ones, or at least not neccessarily so. Think about it this way: what physically relevant substructure stays the same under all these transformations? If I'm right, and they don't neccessarily want minimality here, then there is a good chance that questions a) and b) should also not be answered by describing all the sets with the given property, but instead some (non-minimal unions) which are easy to describe and visualize geometrically. I think the question isn't clear, though, so if you want to play it safe, do both: give the minimal sets as you did, but also characterize some nice unions of these.

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