Uniqueness of the physical measure for impulsive Lorenz semiflows

Determine whether impulsive Lorenz semiflows generated by an impulsive dynamical system (M, X, Σ, φ), where X is a Lorenz flow, φ: Σ → M is a C^2 embedding sufficiently close to the inclusion inc_Σ and satisfies φ(Σ) ⊂ B^+(Σ), admit a unique physical measure. Specifically, establish whether the impulsive semiflow associated with such (M, X, Σ, φ) has exactly one physical probability measure whose basin has positive Lebesgue measure.

Background

The paper proves that, under small impulsive perturbations of a Lorenz flow, the resulting impulsive semiflow admits a finite number of physical measures whose basins cover Lebesgue almost all of the phase space. This extends known results on physical measures for classical Lorenz flows to the impulsive (discontinuous) setting.

For classical Lorenz flows, uniqueness of the physical measure is obtained via the existence of an invariant stable foliation and a transitive quotient map. In the impulsive setting, the semiflow is discontinuous and the associated Poincaré map need not be a skew product or possess a smooth invariant foliation, so these tools are unavailable. The authors note that uniqueness for the impulsive semiflow would follow from uniqueness of the physical measure(s) for the corresponding Poincaré return map, but existing results (e.g., Pesin and Sataev) do not guarantee such uniqueness in this setting.