Timeline for What is the analogue of expansive matrix for automorphisms?
Current License: CC BY-SA 3.0
9 events
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| Oct 23, 2016 at 23:53 | comment | added | YCor | If we consider $\mathbf{Z}^2$ and the matrix $\begin{pmatrix}2 & 1 \\ 1 & 1\end{pmatrix}$, if $p$ is the projection to the eigenspace for the eigenvalue $>1$ (with kernel the other eigenspace) and we define $d(n,m)=|p(m-n)|+1$ for all $m\neq n$, we get a metric which induces the topology, and in addition is translation invariant, and strictly increases the distance of nonzero elements. Do we want to call such a map "expansive"? I doubt it (it preserves the volume). But as far as the OP doesn't say what he/she requires, the question is unclear / too broad. | |
| Oct 23, 2016 at 19:32 | comment | added | David Handelman | Just the existence of some metric yielding the same topology? (This, or anyway a variant of it, is done in actions on Cantor sets, although the definition of expansive is different.) Maybe add to the definition that $\alpha$ is not an isometry, or more extreme, if $x \neq y$, then the inequality is strict. I'm simply asking if this type of definition has been considered. | |
| Oct 23, 2016 at 19:08 | comment | added | YCor | @DavidHandelman A metric with what compatibility conditions? Also I understand from the question that the identity is not considered as an expansive automorphism. | |
| Oct 23, 2016 at 19:08 | comment | added | David Handelman | You can't just use that there exists a metric $d$ such that $d(\alpha x,\alpha y) \geq d(x,y)$? | |
| Oct 23, 2016 at 19:04 | history | edited | David Handelman | CC BY-SA 3.0 |
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| Oct 23, 2016 at 6:29 | comment | added | YCor | You have several options but they should depend on what you require. For instance, if $G$ is discrete, what would it mean to be expansive? And if $G$ is compact? (in both cases, all automorphism preserve the volume.) If it's too extreme, maybe specify what you would call an expansive automorphism of $\mathbb{R}\times\mathbb{Z}$ and $\mathbb{R}\times(\mathbb{R}/\mathbb{Z})$. | |
| Oct 23, 2016 at 2:45 | history | edited | Melody | CC BY-SA 3.0 |
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| Oct 23, 2016 at 1:21 | history | edited | Melody | CC BY-SA 3.0 |
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| Oct 23, 2016 at 1:09 | history | asked | Melody | CC BY-SA 3.0 |