  Misiurewicz[19]introducedtheconceptofpseudo-entropyandproved this quantity
coincides with topological entropy. Richeson et al. [21] obtained the lower
bounded of topological entropy by means of the definition of pseudo-entropy.
This paper aims to generalize the main results obtained by Misiurewicz and
Rich-eson et al. to free semigroup actions. Firstly, the pseudo-entropy is
introduced for free semigroup actions, and it is shown that the pseudo-entropy
coincides with the topological entropy defined by Bufetov [9]. Secondly, these
concepts of the chain recurrence, the chain mixing, the chain recurrence time,
and the chain mixing time for free semigroup actions are introduced, and the
upper bounds of these recurrence times are given. Furthermore, a lower bound of
topological entropy is given by the lower box dimension and the chain mixing
time using the definition of pseudo-entropy for free semigroup actions.
Thirdly, the structure of chain transitive systems for free semigroup actions
is discussed.
