by $A^\infty$. Recall that a subset of a topological space is of *first category* if it can be written as countable union of closed nowhere dense sets. It is *residual* if it is a countable intersection of open and dense sets. A set is *nontrivial* if it contains at least two elements.

Dynamical systems
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Throughout the paper a *dynamical system* means a pair $(X,T)$ where $X$ is a compact metric space and $T\colon X\to X$ is a continuous[^1] map. We say that $(X,T)$ is *invertible* if $T$ is a homeomorphism. We denote a metric on $X$ by $\rho$. We will often identify a dynamical system $(X,T)$ with a map $T\colon X\to X$ alone. We say that $x\in X$ is a *periodic point* for $T$ if $T^k(x)=x$ for some $k\in{\mathbb{N}}$ and we call $k$ a *period* for $x$. We denote the set of all periodic points of $T$ by $\operatorname{Per}(T)$.

Choquet theory
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A nonempty convex compact and metrizable subset $K$ of a locally convex topological vector space is a *Choquet simplex* if every point of $K$ is the barycenter of a unique probability measure supported on the set of extreme points of $K$ (see [@Phelps01]). A *Poulsen simplex* is 