WordNet (r) 3.0 (2006):

sphere of influence
    n 1: the geographical area in which one nation is very
         influential [syn: sphere, sphere of influence]

Moby Thesaurus II by Grady Ward, 1.0:

51 Moby Thesaurus words for "sphere of influence":
   Eisenhower Doctrine, Monroe Doctrine, Nixon Doctrine,
   Truman Doctrine, ambit, appeasement, bailiwick, balance of power,
   brinkmanship, coexistence, colonialism, compromise, constituency,
   containment, detente, deterrence, diplomacy, diplomatic,
   diplomatics, dollar diplomacy, dollar imperialism, expansionism,
   footing, foreign affairs, foreign policy, good-neighbor policy,
   imperialism, internationalism, isolationism, manifest destiny,
   militarism, nationalism, neocolonialism, neutralism, nonresistance,
   open door, open-door policy, orbit, peace offensive,
   peaceful coexistence, preparedness, shirt-sleeve diplomacy,
   shuttle diplomacy, spheres of influence, stamping ground,
   territory, the big stick, tough policy, turf, vantage,
   world politics