We present a theoretical framework for describing electromagnetic kinetic turbulence in a multi-species, magnetized, pressure-anisotropic plasma. Turbulent fluctuations are assumed to be small compared to the mean field, to be spatially anisotropic with respect to it, and to have frequencies small compared to the ion cyclotron frequency. At scales above the ion Larmor radius, the theory reduces to the pressure-anisotropic generalization of kinetic reduced magnetohydrodynamics (KRMHD) formulated by Kunz et al. (2015). At scales at and below the ion Larmor radius, three main objectives are achieved. First, we analyse the linear response of the pressure-anisotropic gyrokinetic system, and show it to be a generalisation of previously explored limits. The effects of pressure anisotropy on the stability and collisionless damping of Alfvenic and compressive fluctuations are highlighted, with attention paid to the spectral location and width of the frequency jump that occurs as Alfven waves transition into kinetic Alfven waves. Secondly, we derive and discuss a general free-energy conservation law, which captures both the KRMHD free-energy conservation at long wavelengths and dual cascades of kinetic Alfven waves and ion entropy at sub-ion-Larmor scales. We show that non-Maxwellian features in the distribution function change the amount of phase mixing and the efficiency of magnetic stresses, and thus influence the partitioning of free energy amongst the cascade channels. Thirdly, a simple model is used to show that pressure anisotropy can cause large variations in the ion-to-electron heating ratio due to the dissipation of Alfvenic turbulence. Our theory provides a foundation for determining how pressure anisotropy affects the turbulent fluctuation spectra, the differential heating of particle species, and the ratio of parallel and perpendicular phase mixing in space and astrophysical plasmas.
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