Athermal disordered systems can exhibit a remarkable response to an applied oscillatory shear: after a relatively few shearing cycles, the system falls into a configuration that had already been visited in a previous cycle. After this point the system repeats its dynamics periodically despite undergoing many particle rearrangements during each cycle. We study the behavior of orbits as we approach the jamming point in simulations of jammed particles subject to oscillatory shear at fixed pressure and zero temperature. As the pressure is lowered, we find that it becomes more common for the system to find periodic states where it takes multiple cycles before returning to a previously visited state. Thus, there is a proliferation of longer periods as the jamming point is approached.Keywords: cyclic shearing, reversibility, memory, absorbing-state phase transition, jammingOscillatory sheared athermal particle packings or suspensions can fall into periodic "absorbing states" [1] in which the system returns to a configuration previously visited during the shearing process at the same point in the cycle. Once it returns to that configuration, the dynamics repeats itself indefinitely. At low densities in the absorbing state, the particles follow the flow without ever making contact with one another so that the system moves back and forth along a flat direction in the energy landscape [2][3][4], and particles return to their original positions after a single shear cycle: T = 1. As the strain amplitude γ t increases beyond some value γ * t , particles can no longer avoid each other and the system undergoes a dynamical "absorbing state" transition from the absorbing phase to a phase in which the system continually visits new configurations. Models [3,[5][6][7][8] have linked this transition to variants of directed percolation [8][9][10], which represents a broad class of non-equilibrium phase transitions [1].Athermal glasses such as Lennard-Jones glasses, by contrast, have an extensive entropy of energy minima that are not flat [11][12][13]. At very small strain amplitudes, they exhibit elastic behavior in which they explore different configurations within the same energy minimum. As γ t increases so that the system can explore more than one minimum, one might expect the system to meander indefinitely around a hopelessly intricate energy landscape as the system is driven in an oscillatory fashion. Yet, remarkably, these systems can fall into absorbing states-they can find their way back to previously visited energy minima even as they undergo multiple particle rearrangements. Thus these systems explore many such minima [14-17] over and over again. Finally, when γ t is increased to γ * t , the system undergoes an absorbing state transition to a phase in which the system never returns to previously visited minima.In this paper we investigate the fate of absorbing states in packings of jammed spheres that can be tuned to the jamming transition, where the system loses rigidity [18,19]. Far above this transition, absorbing states...