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Context: At the end of DS9 Season 2, a Jem'Hadar directly tells Sisko:
"The Dominion will no longer stand by and allow ships from your side to violate our territory."
Question: Isn't the Dominion's complaint totally valid, i.e. that Starfleet ships are invading their territory? Shouldn't the Federation have replied that it was sorry for its screw up?
It is a little fuzzy what, if any, diplomatic channels were opened between the Federation and the Dominion prior to the outbreak of the war. If the Dominion never established exactly what was their territory (rather than perhaps generally claiming the entire quadrant), then the Federation may have felt justified continuing to explore.
The few times that we did see anyone from the Federation trying to find someone from the Dominion to negotiate with it usually ended in a skirmish at best, or the death of many people at worst.
Referencing the same episode you quote in the question, the first official contact between the Federation and the Dominion resulted in the unprovoked loss of a Galaxy-class starship with all hands. Follow that up with "The Search" Part 1 and Part 2 when the Dominion captured the crew of the Defiant and conducted a interactive physiological experiment on the command crew.
Per the female Founder at the end of Part 2, the Dominion is about control and order. They judge solid-form races as threats, regardless of anything else, and to protect themselves they will go to any length to try to control them.
FEMALE: Ironic, isn't it? The hunted now control the destinies of hundreds of other races.
ODO: But why control anyone?
FEMALE: Because what you control can't hurt you. So, many years ago we set ourselves the task of imposing order on a chaotic universe.
KIRA: Is that what you call it? Imposing order? I call it murder.
FEMALE: What you call it is no concern of ours.
ODO: How do you justify the deaths of so many people?
FEMALE: The Solids have always been a threat to us. That's the only the justification we need.
ODO: But these Solids have never harmed you. They travel the galaxy in order to expand their knowledge, just as you once did.
FEMALE: The Solids are nothing like us.
emphasis mine
So while the Federation may have been guilty of violating Dominion space (this is unclear), the Dominion itself hardly gave the Federation any opportunity to apologize for that or broker any sort of right-of-way.
I'd actually say no. Throughout seasons 1 through 3, it's established that the other end of the wormhole isn't in the Dominion's formally controlled territory. The first time the dominion was mentioned was in season 2, in "Rules of Acquisition". That was in season 2. The second time was "Sanctuary", also in season 2. The next time was "Shadowplay".
In "Rules of Acquisition", the Ferengi's knowledge of the Dominion was shaky at best. Additionally, Zek stated that he didn't care about the Dosi, who they were contacting, and they were merely a stepping stone for other Gamma quadrant markets. This is later verified by their contract shown in "The Search, part 1" with their agreement with the Karemma. If the Dosi are a foothold in the dominion economically, rather than critical themselves, this suggests that they're on the outer edge of the Dominion, which is a multi-hour high-warp flight from the wormhole.
In "Sanctuary", the dominion is mentioned exactly once. The Skrreea leader states that their people were first conquered by one race, who were in chain conquered by the dominion. From there, the Skrreea fanned out looking for the eye of the universe, hoping for a new home. The Skrreea only found the wormhole after a massive fan-out operation, with only one vessel finding it initially. "They later contacted the other ships in their fleet." This suggests the dominion's border is fairly far away from the wormhole.
The most interesting matter is in Shadowplay. The guy who founded the holographic colony was from a planet the Dominion took over. However, there's no other mentioning of the dominion in that episode aside from that. This suggests that the Dominion hasn't reached the colony yet, because it hasn't been obviously conquered. This suggests that the colony was beyond the Dominion's border.
None of these planets were within a short distance of the wormhole. On each case, it was a long multi-hour journey at high warp. This suggests that the wormhole wasn't within Dominion territory, but rather fairly far away from their border. Therefore, the Federation's usage of the wormhole wasn't a violation of their borders. This is the Star Trek equivalent of Brazil seeing merchant vessels use the Panama Canal, and calling it "an invasion". So overall, the federation shouldn't apologized for the usage of the wormhole.
https://memory-alpha.fandom.com/wiki/Rules_of_Acquisition_(episode)
In my answer to this question
How far was the wormhole from Dominion territory?1
I use two different lines of reasoning to show that during the third season of DS9 - after the Dominion gave the warming to stay out of the Gamma Quadrant - the nearest Dominion outpost was probably hundreds or thousands of light years from the Gamma Quadrant mouth of the wormhole. About 599.97 to 3,599.82 light years.
The Federation obviously believed that the space for hundreds of light years around the Gamma Quadrant mouth of the wormhole was free and unclaimed and/or belonged to various nearby space traveling societies that were not, repeat NOT part of the Dominion.
Some people have speculated that the Gamma Quadrant aliens met during the first season ere all vassals of the Dominion. But they never even mentioned the Dominion while in the Alpha Quadrant. Wouldn't that be neglecting their duty to the Dominion if they gave the impression that their societies were independent instead of vassals of the Dominion.
If I was a Vorta in charge of those worlds and learned their representatives had never even mentioned their Dominion vassal status while dealing with a major space power, I would be inclined to have those representatives arrested and executed for lese-majesty and treason against the Dominion.
The popular idea that those first season Gamma quadrant aliens could have belonged to vassal planets of the Dominion seems crazy to me.
Thus the Federation government believed that they had every right to travel to the Gamma Quadrant and explore parts of the Gamma Quadrant that did not belong to the Dominion.
But it is quite possible that later the Federation government apologized to the Federation citizens for annoying the Dominion by going into the Gamma quadrant and getting into a bloody war with the Dominion.
