are closed, invariant, and the restriction of $T$ to each of them is topologically transitive. Furthermore, Bowen proved that if $\Lambda$ is a basic set for $T$, then $\Lambda$ can be decomposed into disjoint closed sets $\Lambda_1,\ldots,\Lambda_m$ such that $T(\Lambda_i)=\Lambda_{(i+1)\bmod m}$ and $T^m|\Lambda_i$ has the periodic specification property and that is how this property entered into mathematics. Some authors call $\Lambda_1,\ldots,\Lambda_m$ *elementary sets*.

Let $\nu\colon {\mathbb{N}}\to{\mathbb{N}}$ be any function. A family of orbit segments $\xi=\{T^{[a_j,b_j]}(x_j)\}_{j=1}^n$ is a *$\nu$-spaced specification* if $a_i-b_{i-1} \ge  \nu(b_i-a_i+1) $ for $2\le i \le n$. Given a *constant* $N\in{\mathbb{N}}$ by an *$N$-spaced specification* we mean a $\nu$-spaced specification where $\nu$ is the *constant function* $\nu(n)= N$ for all $n\in{\mathbb{N}}$.

We say that a specification $\xi=\{T^{[a_j,b_j]}(x_j)\}_{j=1}^n$ is ${\varepsilon}$-traced by $y\in X$ if $$\rho(T^k(y),T^k(x_i)) \le {\varepsilon}\quad\text{for } a_i\le k \le b_i\text{ and }1\le i \le n.$$

\[def:spec\] We say that $(X,T)$ has the *specification property* if for any ${\varepsilon}>0$ there is a constant $N = N({\varepsilon})\in{\mathbb{N}}$ such that any $N$-spaced specification $\xi=\{T^{[a_j,b_j]}(x_j)\}_{j=1}^n$ is ${\varepsilon}$-traced by some $y\in X$. If additionally, $y$ can be chosen in such a way that $T^{b_n-a_0+N}(y)=y$ then $(X,T)$ has the *periodic specification property*.

Some authors consider a weaker notion, which we propose to name 