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The Plaque Inverse Limit of a branched covering self-map of a Riemann surfacewas introduced and studied in \cite{CCG}. A point $x$ of P.I.L. was calledregular if P.I.L. has the natural Riemann Surface structure at $x$ and wascalled irregular otherwise. The notion of the signature $sign(x,c)$ of $x$ withrespect to a critical point $c$, which was shown to be a local invariant ofP.I.L. was introduced and developed. It was shown that $sign(x,c)$ isnontrivial for some critical points $c$ if and only if $x$ is an irregularpoint. 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 somepath-connected component of that neighborhood into an uncountable number ofpath-connected components. Finally, various signatures, including signatures ofthe invariant lifts of super-attracting and attracting cycles and certainsignatures of the invariant lift of a parabolic cycle, were computed. All thesesignatures had a maximal element. In this work we show that the local topologyof 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 aproperty, 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-connectedcomponents, 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 thesignature of the invariant lift of a parabolic cycle with respect to a certainrecurrent 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 paraboliccycles, have no maximal element with respect to some recurrent critical point.

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