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Inspired by the renewed experimental activities on $p$-wave resonantlyinteracting atomic Fermi gases, we theoretically investigate some experimentalobservables of such systems at zero temperature in two dimensions, using bothmean-field theory and Gaussian pair fluctuation theory. These observablesinclude the two $p$-wave contact parameters and the breathing mode frequency,which can be readily measured in current cold-atom setups with $^{40}$K and$^{6}$Li atoms. We find that the many-body component of the two contactparameters exhibits a pronounced peak slightly above the resonance andconsequently leads to a dip in the breathing mode frequency. In the resonancelimit, we discuss the dependence of the equation of state and the breathingmode frequency on the dimensionless effective range of the interaction,$k_{F}R_{p}\ll1$, where $k_{F}$ is the Fermi wavevector and $R_{p}$ is theeffective range. The breathing mode frequency $\omega_{B}$ deviates from thescale-invariant prediction of $\omega_{c}=2\omega_{0}$, where $\omega_{0}$ isthe trapping frequency of the harmonic potential. This frequency shift iscaused by the necessary existence of the effective range. In the small rangelimit, we predict that the mode frequency deviation at the leading order isgiven by, $\delta\omega_{B}\simeq-(\omega_{0}/4)\ln^{-1}(k_{F}R_{p})$.

Resonantly interacting $p$-wave Fermi superfluid in two dimensions: Tan's contact and breathing mode