Distinguishing super-Nyquist frequencies via their temporal variation in $\gamma$ Doradus stars from continuous photometry

As mixed with real pulsations, the reflection of super-Nyquist frequencies (SNFs) pose a threat to asteroseismic properties. Although SNFs have been studied in several pulsating stars, a systematic survey remains scarcely explored. Here we propose a method to identify SNFs from Kepler and TESS photometry by characterizing their periodic frequency modulations using a sliding Fourier transform. After analyzing long cadence photometry in the Kepler legacy, we have identified 304 SNFs in 56 stars from 45607 frequencies in $\sim600$ $\gamma$ Doradus stars, corresponding to a fraction of approximately $0.67\%$ and $9.2\%$, respectively. Most SNFs are detected in the frequency range of pressure mode over 120 $\mu$Hz and the fraction of SNF detection increases as frequency up to $\sim7\%$. We barely found two potential SNFs mixed with gravity modes in two $\gamma$ Doradus stars. These findings indicate that SNFs have a negligible impact on the global seismic properties, such as those derived from period spacing in $\gamma$ Doradus stars. However, we stress that SNFs must be carefully and systematically examined by this method in other pulsating stars, particularly $\delta$ Scuti and hot B subdwarf stars, to establish a solid foundation for precise asteroseismolgy of various types of pulsators.