sets has positive topological entropy.

The topological $K$-systems are also known as systems with *uniform positive entropy of all orders*. In this nomenclature Blanchard’s *uniform positive entropy* is the *uniform positive entropy of order $2$*. Every minimal topological $K$-system is mixing [@HSY05]. Huang and Ye [@HY06 Theorem 7.4] observed that topological $K$-systems have a kind of a very weak specification property.

Here we only mention an easy part of this connection (cf. [@HY06], Theorem 7.4). It is easy to see that if a surjective system $(X,T)$ has the specification property, then for any nonempty open sets $U_1,\ldots,U_k\subset X$ there is an $N$ such that for any $n\in{\mathbb{N}}$ and ${\varphi}\colon {\left\{0,\ldots,n\right\}}\to {\left\{1,\ldots,k\right\}}$ there is a point $z$ satisfying $T^{iN}(z)\in U_{{\varphi}(i)}$ for $i=0,\ldots,n$. This immediately gives the following (cf. Proposition 21.6 in [@DGS] and Proposition 4.3 in [@Bowen71]).

If a surjective system $(X,T)$ has the specification property, then it is a topological $K$-system.

The next result is a consequence for $d=1$ of Theorem B in [@EKW] (Eizenberg, Kifer and Weiss stated it for $\mathbb{Z}^d$ actions). Theorem B in [@EKW] asserts that if $(X,T)$ is an invertible dynamical system with the specification property and $\mu$ is a $T$-invariant probability measure such that the 