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We report on quantum capacitance measurements of high quality, graphite- andhexagonal boron nitride encapsulated Bernal stacked trilayer graphene devices.At zero applied magnetic field, we observe a number of electron density- andelectrical displacement-tuned features in the electronic compressibilityassociated with changes in Fermi surface topology. At high displacement fieldand low density, strong trigonal warping gives rise to emergent Dirac gulliescentered near the corners of the hexagonal Brillouin and related by three foldrotation symmetry. At low magnetic fields of $B=1.25$~T, the gullies manifestas a change in the degeneracy of the Landau levels from two to three. Weakincompressible states are also observed at integer filling within thesetriplets Landau levels, which a Hartree-Fock analysis indicates are associatedwith Coulomb-driven nematic phases that spontaneously break rotation symmetry.

Emergent Dirac gullies and gully-symmetry breaking quantum Hall states in ABA trilayer graphene