henceforth concentrate on dynamical systems $(X,T)$ given by an **onto map on a nontrivial space**. Note that some authors (see for example [@Yamamoto09]) use a slightly different definition of the specification property which implies surjectivity and for onto maps is equivalent to Definition \[def:spec\]. For the sake of completeness we recall that a dynamical system $(X,T)$ has the specification property as defined in [@Yamamoto09] if for any ${\varepsilon}>0$ there is an integer $M_{\varepsilon}$ such that for any $k\ge 1$ and $k$ points $x_1,\ldots,x_k\in X$ and for any sequence of integers $0\le a_1\le b_1< a_2\le b_2<\ldots<a_k\le b_k$ with $a_{i}-b_{i-1}\ge M_{\varepsilon}$ for $2\le i\le k$, there is an $x\in X$ with $\rho(T^{a_i+j}(x),T^j(x_i))\le {\varepsilon}$ for $0\le j\le b_i-a_i$ and $0\le i\le k$. Note that periodic specification is called *strong specification* in [@Yamamoto09].

Observe that a dynamical system $(X,T)$ is topologically transitive if and only if for every $x_0,\ldots, x_k\in X$ and $n_0,\ldots, n_k\in\mathbb N_0$ there exist $m_1,\ldots,m_k\in{\mathbb{N}}$ such that $$\bigcap_{j=0}^k T^{-\sum_{i=1}^{j}(n_{i-1}+m_i)}B_{n_j}(x_j,{\varepsilon})\neq\emptyset.$$ Clearly, each $m_j$ depends on all points $x_i$, all $n_i$ and ${\varepsilon}$. Therefore the specification property can be considered as a uniform version of transitivity, which allows us to pick all $m_j$ equal to a constant depending only on ${\varepsilon}$.

The following 