# How slowly can the critical points of the Fibonacci terms grow?

Define the Fibonacci terms $t_{n}$ for all $n\geq 1$ by letting $t_{1}(x,y)=y,t_{2}(x,y)=x$, and $t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$ for all $n$. The Fibonacci terms are used in order to approximate the composition operation on an algebra of elementary embeddings with application.

Let $\mathcal{E}_{\lambda}$ denote the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$, and let $\mathcal{E}_{\lambda}^{+}$ denote the set of all non-identity elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. Recall that $\mathcal{E}_{\lambda}$ is endowed with a self-distributive operation $*$ called application defined by $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. Let $\equiv^{\gamma}$ denote the congruence on $\mathcal{E}_{\lambda}$ where $j\equiv^{\gamma}k$ precisely when $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for all $x\in V_{\gamma}$.

Suppose that $j,k\in\mathcal{E}^{+}_{\lambda}$ and $\mathrm{crit}(k)\geq\mathrm{crit}(j)$. Then what are some lower bounds of the necessarily finite cardinality $$|\langle j,k\rangle\equiv^{\mathrm{crit}(t_{2n+1}(j,k))}|?$$ What are some lower bounds of the necessarily finite cardinality $$|\{\mathrm{crit}(l)|l\in\langle j,k\rangle,\mathrm{crit}(l)\leq\mathrm{crit}(t_{2n+1}(j,k))\}|?$$

• Aha so $y,x,yx,xyx,yxxyx,\cdots$ – Aaron Meyerowitz Oct 21 '17 at 16:09
• @AaronMeyerowitz If I understand the setup correctly, the $*$ operation is not associative, hence the positions of brackets in the terms are significant. – Emil Jeřábek Oct 26 '17 at 9:36
• All right, I've seen just about enough. Closers: this question seems reasonable and I am practically certain to reopen it if it gets closed. A judgment to close should be supported by some knowledge of the area. Joseph: quit fanning the flames by calling people trolls. @GerryMyerson: while Joseph could have calmly answered your question, he is right that this is standard notation. In ZFC, the recursive definition of $V_\alpha$ for ordinals $\alpha$ (Jech, p. 64) is that $V_0 = \emptyset, V_{\alpha+1} = P(V_\alpha)$, and $V_\alpha = \bigcup_{\beta < \alpha} V_\beta$ for limit ordinals $\alpha$. – Todd Trimble Oct 26 '17 at 13:43
• @Todd Joseph did answer both Aaron and Gerry. For instance, Aaron's second comment was in response to Joseph's reply, and Joseph replied to it as well. (And, after a few days, he deleted his comments.) – Andrés E. Caicedo Oct 26 '17 at 14:10
• @AndrésE.Caicedo Yes, you're right. But it's hard to tell whether those who asked ever saw his replies, since deleted comments are invisible to all except site moderators and community managers. – Todd Trimble Oct 26 '17 at 14:24