measure generated by $x$. It is *quasiregular* for $T$ if there exists $\mu\in\operatorname{\mathcal{M}_T}(X)$ such that $x$ is generic for $\mu$.

Measure center
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An open set $U\subset X$ is *universally null* in a dynamical system $(X,T)$ if $\mu(U)=0$ for every $\mu\in\operatorname{\mathcal{M}_T}(X)$. The *measure center* of $(X,T)$ is the complement of the union of all universally null sets, or equivalently, it is the smallest closed subset $C$ of $X$ such that $\mu(C)=1$ for every $\mu\in\operatorname{\mathcal{M}_T}(X)$. Another characterization of the measure center uses ideas of Birkhoff and Hilmy. Birkhoff introduced the *probability of sojourn*, defined for $x\in X$ and $U\subset X$ as $$p(x,U)=\limsup_{N\to\infty}\frac{1}{N} \left|{\left\{0\le n<N:T^n(x)\in U\right\}}\right|.$$ Hilmy [@Hilmy36] defined the *minimal center of attraction* of a point $x\in X$ as $$I(x)={\left\{y\in X : p(x,U)>0 \text{ for any neighborhood } U \text{ of }y\right\}}.$$ It can be proved (see [@Sigmund77]) that the measure center is the smallest closed set containing the minimal center of attraction of every point $x\in X$. If the minimal points are dense in $X$ then the measure center is the whole space, but without density of minimal points no specification-like property we consider can guarantee that.

Entropy
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Measure-theoretic and topological entropies are among the most important invariants in 