"How ya doin?" v. "How ya' doin'?" and so on.
There's... a debate.
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It doesn't make sense to put an apostrophe after "ya", because no letters have been omitted. And that's what apostrophe indicates — it isn't a general clue that a word is shortened in some way.
In "ya", the "ou" vowel has been replaced with "a". We don't have punctuation to indicate that, so we just write it.
This is also generally the case where a replacement slang/informal word is missing letters, but others have changed. When this happens, we usually just transcribe the sounds rather than using an apostrophe. For example, we write "gimme" for the hurried expression which means "give me". We don't write "gi'me", even though that might make logical sense. And, we write "gonna" for "going to", rather than.... "go'na", I guess. (Yeah, not that.)
I would suggest that at least in some dialects, the phrase "are you" is (informally) spoken as something between "ee-ah" and "eh-ya", which is distinct from "ya" meaning "you"; I would suggest
'ya (with the apostrophe) as an orthography for shortened form of "are you", since any other notation for that initial vowel sound would be apt to cause confusion. In many cases, context would distinguish between the usages even without a difference in spelling, but I would regard the following as different questions, with somewhat different pronunciations:
The former question would be asking whether someone is confused about their location; the latter would ask whether they had been vanquished.
I would thus notate the question
How 'ya doin' with the apostrophe as shown, pronounced as "Howie adieuing"; if
ya were used to simply mean "you", as in "I don't know how ya do it", I would omit the apostrophe.