A large fraction of stars is found to be part of binary or higher-order multiple systems. The ubiquity of planets found around single stars raises the question if and how planets in binary systems may form. Protoplanetary disks are the birthplaces of planets, and their characterization is crucial in order to understand the planet formation process. Our aim is to characterize the morphology of the GG Tau A disk, one of the largest and most massive circumbinary disks, and trace evidence for binary-disk interactions. We obtained observations in polarized scattered light of GG Tau A using the SPHERE/IRDIS instrument in the H-band filter. We analyze the observed disk morphology and substructures. We run 2D hydrodynamical models simulating the evolution of the circumbinary ring over the lifetime of the disk. The disk, as well as the cavity and the inner region are highly structured with several shadowed regions, spiral structures, and streamer-like filaments, some of them detected for the first time. The streamer-like filaments appear to connect the outer ring with the northern arc. Their azimuthal spacing suggests that they may be generated by periodic perturbations by the binary, tearing off material from the inner edge of the outer disk once during each orbit. By comparing observations to hydrodynamical simulations we find that the main features, in particular the gap size, as well as the spiral and streamer filaments, can be qualitatively explained by the gravitational interactions of a binary with semi-major axis of $sim$35 au on an orbit coplanar with the circumbinary ring.
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