  Introducing the notions of (inner) $\sigma$-derivation, (inner)
$\sigma$-endomorphism and one-parameter group of $\sigma$-endomorphisms
($\sigma$-dynamics) on a Banach algebra, we correspond to each
$\sigma$-dynamics a $\sigma$-derivation named as its $\sigma$-infinitesimal
generator. We show that the $\sigma$-infinitesimal generator of a
$\sigma$-dynamics of inner $\sigma$-endomorphisms is an inner
$\sigma$-derivation and deal with the converse. We also establish a nice
generalized Leibniz formula and extend the Kleinenckr--Sirokov theorem for
$\sigma$-derivations under certain conditions.
