Let { A 1 , … , A K } ⊂ C d × d be arbitrary K matrices, where K and d both ⩾2. For any 0 < Δ < ∞ , we denote by L Δ pc ( R + , K ) the set of all switching sequences u = ( λ . , t . ) : N → { 1 , … , K } × R + satisfying t j − t j − 1 ⩽ Δ and 0 = : t 0 < t 1 < ⋯ < t j − 1 < t j < ⋯ with t j → + ∞ . Differently from the classical weak-⁎ topology and L 1 -norm, we equip L Δ pc ( R + , K ) with the topology so that the “one-sided Markov-type shift” ϑ + : L Δ pc ( R + , K ) → L Δ pc ( R + , K ) , defined by u = ( λ j , t j ) j = 1 + ∞ ↦ ϑ + ( u ) = ( λ j + 1 , t j + 1 − t 1 ) j = 1 + ∞ , is continuous, which is different from and simpler than the classical continuous-time “translation”. We study the stability of the linear switched dynamics ( A): x ˙ ( t ) = A u ( t ) x ( t ) , x ( 0 ) ∈ C d and t > 0 where u ( t ) ≡ λ j if t j − 1 < t ⩽ t j , for any u ∈ L Δ pc ( R + , K ) . By introducing the concept “weakly Birkhoff recurrent switching signal”, we show that if, under some norm ‖ ⋅ ‖ , the principal matrix Φ u ( t ) of ( A) satisfies ‖ Φ u ( t ) ‖ ⩽ 1 for all u ∈ L Δ pc ( R + , K ) and t > 0 , then for any ϑ + -ergodic probability P on L Δ pc ( R + , K ) , either lim j → + ∞ 1 j log ‖ Φ u ( t j ) ‖ < 0 for P -a.s. u = ( λ j , t j ) j = 1 + ∞ ; or ‖ Φ ϑ + j ( u ) ( t j + k − t j ) ‖ = 1 ∀ k , j ⩾ 0 for P -a.s. u = ( λ j , t j ) j = 1 + ∞ . Some applications are presented, including: (i) equivalence of various stabilities; (ii) almost sure exponential stability of periodically switched stable systems; (iii) partial stability; and (iv) how to approach arbitrarily the stable manifold by that of periodically switched signals and how to select a stable switching signal for any initial data.