For a general non-Hermitian Floquet topological system, the interplay between periodic driving and non-Hermiticity not only complicates and diversifies the definition of the topological invariant but also makes it difficult to establish a direct correspondence between the topological invariant and dynamical observable quantities. In this work, we demonstrate that the winding number of a one-dimensional non-Hermitian Floquet system can be defined via the subspace of the effective Hamiltonian in both the frequency and time domains. More importantly, the topological winding number can be fully determined by measuring the dynamic winding number, which is defined in terms of the time-averaged stroboscopic spin texture, regardless of the initial state, driving frequency, and $\mathcal{P}\mathcal{T}$-symmetry. It is found that periodic driving can induce band inversion and create new band-gap closing points in the subspace of the effective Hamiltonian. These closing points constitute the origin of spontaneous $\mathcal{P}\mathcal{T}$-symmetry breaking. In particular, the $\mathcal{P}\mathcal{T}$-symmetry near the closing point induced by band inversion is more robust against perturbations of the gain or loss parameter than that associated with parameter renormalization. Our work not only clarifies the definition of the topological invariant for non-Hermitian Floquet systems in both the frequency and time domains, but it also provides a simpler dynamical approach for detecting non-Hermitian Floquet topological phases.

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• Cold gases in optical lattices
• PT-symmetric quantum mechanics
• Floquet systems
• Non-Hermitian systems