The present invention relates to the technical field of energy dissipation and shock absorption buildings, and particularly relates to a self-recovering energy dissipation support with a shape memory alloy damper. The self-recovering energy dissipation support includes a core shape memory alloy damper and cross-shaped steel columns, wherein the shape memory alloy damper includes two sets of inner and outer sleeves. A sliding groove is arranged between the inner sleeve and the outer sleeve, so that the inner sleeve and the outer sleeve can slide relative to each other along a track. The two sets of inner sleeves are connected through pre-stretched shape memory alloy ribs I. The inner sleeves and the outer sleeves are connected through pre-stretched shape memory alloy ribs II. An outer end plate of the shape memory alloy damper is connected with the cross-shaped steel columns.