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Can there exist non-trivial elementary embeddings $j,k:V_{\lambda+1}\rightarrow V_{\lambda+1}$ along with a strictly increasing function $r:\omega\rightarrow\omega$ such that $j^{r(2n)}(\mathrm{crit}(j))>k^{r(2n)}(\mathrm{crit}(k))$ and $j^{r(2n+1)}(\mathrm{crit}(j))<k^{r(2n+1)}(\mathrm{crit}(k))$ for all $n$?

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One can easily arrange this for embeddings $V_\lambda\to V_\lambda$, and I think the same method works for $V_{\lambda+1}$.

To do it, start with any $j:V_\lambda\to V_\lambda$. The various compositions $j\circ j\circ\cdots\circ j$ with itself push up the target $j(\kappa)$ as high as desired. And the application $j\cdot j\cdots j$ move the critical point as high as desired. By combining these operations, one can arrange for embeddings with a given critical point $\kappa_n$ and sending it as high as desired in $\lambda$.

By taking finite compositions of such embeddings, one can arrange to fulfill your desired pattern for any finitely many steps.

And now, the point is that these resulting embeddings can be found so as to agree more and more on the rank initial segments of $V_\lambda$. That is, the later compositions only change things increasingly high up in $V_\lambda$.

Therefore, we can take the limit of the embeddings, to produce elementary embeddings that exhibit your desired pattern all the way.

This will produce embeddings $j,k:V_\lambda\to V_\lambda$ with your interleaving phenomenon. Indeed, one can make it happen with basically any kind of interleaving pattern.

It seems to me that the argument will also work with embeddings on $V_{\lambda+1}$, but I need to think a little more about whether the limit of embeddings is still elementary on $V_{\lambda+1}$.

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    $\begingroup$ The 1997 paper "Implications between strong large cardinal axioms" in Theorem 1 states that some direct limit (and hence some topological limit) of elementary embeddings from $V_{\lambda+1}$ to $V_{\lambda+1}$ is not necessarily even a $\Sigma_{1}^{1}$ embedding while the inverse limit of $V_{\lambda+1}$ elementary embeddings is always elementary on $V_{\lambda+1}$. I probably should have checked this paper once again before asking because I was caught off guard by the fact that inverse limits preserve $V_{\lambda+1}$ elementarity while I knew that topological limits do not. $\endgroup$ – Joseph Van Name Jan 18 at 23:03
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    $\begingroup$ For the readers, the direct limits are the elementary embeddings $\lim_{n} j_{n}\circ...\circ j_{0}$ such that $j_{n}\circ...\circ j_{0}(\alpha)$ eventually constant for all $\alpha$. The inverse limits are the elementary embeddings $\lim_{n}j_{0}\circ...\circ j_{n}$ where $\mathrm{crit}(j_{0})<...<\mathrm{crit}(j_{n})$ and $(\mathrm{crit}(j_{n}))_{n}$ is cofinal in $\lambda$. Inverse limits should do the trick in producing $V_{\lambda+1}$-elementary embeddings with interweaving growth rates. $\endgroup$ – Joseph Van Name Jan 18 at 23:08
  • $\begingroup$ Yes, I agree, the construction idea I mentioned is going to be an inverse limit. $\endgroup$ – Joel David Hamkins Jan 19 at 14:12

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