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The three men who rule Rome after Caesar's death are Antony, Lepidus, and Octavius Caesar.
Antony was a friend of Julius Caesar's and was dismayed over his death; however, he was smart enough not to tell the conspirators that immediately after the assassination! He knew that he needed to bide his time and use his funeral speech for Caesar as the way to turn the people against those that murdered Caesar. Antony is smart, loyal, and not given to rash acts of passionate nature - he uses his common sense.
Lepidus is actually a rather minor character in the play's action. According to Octavius, Lepidus is a "tried and valiant soldier" (4.1), to which Antony replies, "So is my horse." Antony expresses concern over whether Lepidus is really up to the task of ruling Rome:
"This is a slight, unmeritable man,
Meet to be sent on errands." (4.1)
Octavius Caesar is the heir to Julius Caesar (his great-nephew)because JC did not have children. Octavius is very opinionated, as mentioned above in dealing with Antony's thoughts on Lepidus. In addition, Octavius decided to follow his own military strategies at the Battle of Philippi, rather than take orders from Antony, which lets us know that these two are not going to rule Rome peaceably together.