Captain Braxton aboard Aeon.
Captain Braxton was a Starfleet officer from the 29th century. Braxton was first encountered by the USS Voyager in 2373, when his timeship, the Aeon, arrived in the Delta Quadrant. Braxton claimed that Voyager had been responsible for the destruction of the Sol system in the 29th century, and had traveled back to prevent that occurrence — by destroying Voyager. Voyager was able to disrupt Braxton's systems, however, and the Aeon was pulled into an unstable temporal rift, and crash-landed on Earth in 1967. Braxton was able to beam away before impact, but was unable to retreive the timeship, which fell into the hands of Henry Starling.
The homeless Braxton, having aged 30 years.
When Voyager arrived in Earth orbit in 1996, they discovered Braxton living homeless on the streets of Los Angeles. Poor treatment at the hands of the 20th century authorities had left him paranoid and aggressive — blaming Voyager for his predicament. He was able to provide vital information to Captain Janeway which helped resolve the crisis. The fate of this version of Braxton after the resolution of the temporal paradox is unclear. An alternate version of Braxton who never experienced these events traveled to 1996 to return Voyager to the Delta Quadrant. (VOY: "Future's End", "Future's End, Part II")
Voyager next encountered Captain Braxton in 2375, when the ship was mysteriously destroyed by a temporal disruptor. Seven of Nine was recovered from the doomed vessel by Braxton, now commanding the USS Relativity. Braxton sent Seven back to 2371 and 2372 in an attempt to track down and apprehend the temporal criminal responsible. It was discovered that Braxton himself was responsible for the incident, having succumbed to temporal psychosis at an undetermined future point, and vowed to avenge himself on Voyager — citing his exile on 20th century Earth as his motive. At least two versions of this Braxton were recovered by the Relativity, and Braxton's first officer, Ducane, stated that these two — plus the unaffected Braxton, who had been arrested for crimes he was going to commit — would be temporally integrated prior to standing trial. The temporal disruptor was recovered, and the destruction of Voyager was erased from history.
Captain Braxton aboard the Relativity.
Braxton's attempt to sabotage and destroy Voyager stemmed from an event where Voyager caused Braxton to be relieved of command, forced back into rehabilitation and retirement. It is possible that the cause of that was indeed his own actions in trying to destroy Voyager (which would be an example of the Pogo paradox). It is very possible that by going back in time and trying to destroy Voyager, his past self was implicated and arrested for crimes he was going to commit, reintegrated, and rehabilitated; then, suffering from temporal psychosis, sought to destroy the ship (Voyager) and the person (Janeway) that had caused him so much suffering, thus creating a cycle in which he recreated the exact same circumstances in which he was punished for. It is also worth noting that in every instance in which Braxton meets havoc involving Voyager that we witness is his own fault. When Braxton goes back to destroy Voyager because its debris was found in the future, it ended up being his own ship that would have destroyed it, which would not have happened if not for his attempt to destroy Voyager. (VOY: "Relativity")
Background
Allan G. Royal portrayed Braxton in "Future's End," with Bruce McGill taking the role in "Relativity."
It is unclear exactly how the later Braxton remembers being trapped on Earth. However, given the continued existence of the Doctor's emitter and the fact that Voyager continues to orbit Earth in 1996 rather than revert to the Delta Quadrant, it seems likely that the Braxton who was trapped for 30 years also continued to exist. One can speculate that he was picked up and underwent temporal integration with the second Braxton before returning to duty.