Plaque Inverse Limit of a Dynamical System - Dynamics, Signatures and Local Topology

The Plaque Inverse Limit of a branched covering self-map of a Riemann surface was introduced and studied in \cite{CCG}. A point $x$ of P.I.L. was called regular if P.I.L. has the natural Riemann Surface structure at $x$ and was called irregular otherwise. The notion of the signature $sign(x,c)$ of $x$ with respect to a critical point $c$, which was shown to be a local invariant of P.I.L. was introduced and developed. It was shown that $sign(x,c)$ is nontrivial for some critical points $c$ if and only if $x$ is an irregular point. It was shown that the local topology of P.I.L. at an irregular point $x$ has a property, that removing $x$ from any its neighborhood breaks some path-connected component of that neighborhood into an uncountable number of path-connected components. Finally, various signatures, including signatures of the invariant lifts of super-attracting and attracting cycles and certain signatures of the invariant lift of a parabolic cycle, were computed. All these signatures had a maximal element. In this work we show that the local topology of P.I.L. at irregular points with different types of signatures is different. Namely, we prove that the local topology at an irregular point $x$ has a property, that for any neighborhood $V$ of $x$ and for some point $y\ne x$ in $V$, the open set $V-\{y\}$ consists of uncountable number of path-connected components, if and only if the signature $sign(x,c)$, for some critical point $c$, has no maximal element. Next, for a polynomial functions, we compute the signature of the invariant lift of a parabolic cycle with respect to a certain recurrent critical point. This signature, unlike the cases studied in \cite{CCG}, has no maximal element. We show that all other irregular points, except the invariant lifts of super-attracting, attracting, and parabolic cycles, have no maximal element with respect to some recurrent critical point.