It seems like, especially in fast growing empires like those of Genghis Khan and Alexander, that the control of the emperor couldn't have extended very far past where his army happened to be, and they couldn't have had much time to build the infrastructure of government in all the places they conquered, and that various areas were constantly in rebellion. How real are the boundaries you see on maps in history books? Would people who lived in all those places have recognized that they lived in those empires? Where there any formal agreements in place recognizing those borders?
For Rome at least there were informal recognition of borders
Sallust, in the Jugurtha, describes one formal border. During a report of a war between Carthage and Cyrene he writes,
They therefore called a truce and agreed that on a given day envoys should set out from each city and that the place where they met should be regarded as the common frontier of the two peoples.
There are problems. Aside from this reference, I don't know anyone ever claimed such a war happened. Also the story is odd in that it ends with burial alive of the Carthaginians as a guarantee of the border, which smells fishy. Sallust mentions a cult to the ambassadors on the spot, so this very well could be etiology more than anything else.
More commonly physical obstacles be borders. In this same story Sallust explains the need for the strange ritual because
There was neither river nor hill to mark the frontiers.
Other such boundaries that come immediately to mind are the Ebro in Spain (past which Hannibal was not to advance), the Rubicon, the Rhine, and the Alps.
These, like I said, were informal borders. In other contexts it didn't matter so much. Caesar's provinces were The Province (now southern France), Cis-Alpine Gaul, and Illyria. He had no business in Gaul proper, and certainly not in Germany. Geographical boundaries on the outskirts of his provinces didn't seem to matter much to him.