Let $\{ (p_n, \mathcal{ D}_n, L_n)\}$ be a sequence of Hadamard triples on $\mathbb{ R}$. Suppose that the associated Cantor-Moran measure $$ \mu_{ \{ p_n,\mathcal{ D}_n\}}=\delta_{ p_1^{ -1}\mathcal{ D}_1}\ast\delta_{ (p_2p_1)^{ -1}\mathcal{ D}_2}\ast\cdots, $$ where $\sup_n\{ |p_n^{ -1}d|:d\in \mathcal{ D}_n\}<\infty$ and $\sup\#\mathcal{ D}_n<\infty$. It has been observed that the spectrality of $\mu_{ \{ p_n,D_n\}}$ is determined by equi-positivity. A significant problem is what kind of Moran measures can satisfy this property. In this paper, we introduce the conception of \textit{ Double Points Condition Set} (\textit{ DPCS}) to characterize the equi-positivity equivalently. As applications of our characterization, we show that all singularly continuous Cantor-Moran measures are spectral. For the absolutely continuous case, we study Fuglede's Conjecture on Cantor-Moran set. We show that the equi-positivity of $\mu_{ \{ p_n,D_n\}}$ implies the tiling of its support, and the reverse direction holds under certain conditions.
Categories: math.FA