The universal bifurcation diagram conjecture

Let $R:X\to X$ be the renormalization operator from the theorem of Multiverse chaos. A bifurcation diagram is the diagram recording the bifurcations occurring along a homotopy of dynamical systems.

Universal bifurcation diagram conjecture. There exists a homotopy

$$R_t:X\to X,\qquad t\in[0,1],$$

such that $R_1=R$, while $R_0$ has simple dynamics, for example a finite number of periodic orbits. In particular, the bifurcation diagram $\frak{R}$ of this curve is well-defined. Moreover, if $g_t:S\to S$, $t\in[0,1]$, is a $C^1$-isotopy of a closed, smooth manifold $S$ of dimension $d>1$, and $\Gamma_g$ denotes the bifurcation diagram of $g_t$, then $\Gamma_g$ can be embedded as a subgraph of $\frak{R}$.

This conjecture proposes a universal bifurcation analogue of the theorem of Multiverse chaos: a single renormalization homotopy should contain, as subgraphs, the bifurcation diagrams of all such isotopies.