When Pippin looked into the Palantir of Orthanc, why did Sauron believe Pippin was the ring bearer?
I'm just a couple of chapters into The Return of the King, and it isn't clear to me from the movie, so I'm not sure why Sauron thought Pippin had the One Ring just because Pippin picked up the Palantir of Orthanc and looked into it (and had his Palantir experience)? Why did Sauron conclude that Pippin had the One Ring?
The palantír Pippin uses is the one Sauron knows Saruman has in Orthanc. Sauron is also aware that a Hobbit is the Ringbearer. So when a Hobbit looks into the stone that Sauron believes is in Orthanc, he immediately assumes that Saruman has captured the Ringbearer and (in all likelihood) now holds the Ring. And let’s not forget that the book mentions several times that Sauron is full of fear, which probably clouds his judgement.
In the book, Pippin also passes on the message Sauron gave to him, which was “Tell Saruman that this is not for him!” Sauron then dispatched a Nazgûl to retrieve the ‘Ring’.
Long answer short: he put 2 and 2 together and came up with 5.
Sauron knew that a halfling (aka Hobbit) had the ring, and a halfling touched the Palantir. Pippin didn't reveal any information (especially concerning Sam and Frodo's quest), so I'd guess it was just an assumption on Sauron's part. After all, how many hobbits would even have access to or be near a Palantir? Hobbits weren't exactly known for straying too far from Hobbiton. Thus, Sauron assumed that the ring-bearer was Pippin.
As discussed in this related question, Pippin was probably the 'dainty' that Sauron wanted handed over. He could then interrogate Pippin in his dark tower, taking over Saruman's line of investigation into the whereabouts of the One Ring.
I also think Sauron knew the Hobbits were heading to Isengard. The fellowship breaks on Feb 26th and Pippin holds the Palantir on March 4/5. My point is that Sauron had orc spies along the Anduin that surely were aware that some of the fellowship started off toward Isendard. They would have seen the dead orcs which Boromir killed. Then 8/9 days later a hobbit at Orthanc (see other answers). He would have put it together by now. Even if the Mordor orcs that were traveling with the Uruk-Hai back to Isengard were all killed**, news would have made it back to Sauron from his spies along the river.
** I remember reading a point where all of the Mordor orcs with the Uruk-hai are killed and its explicitly pointed out that Sauron would never find out that the hobbits (Merry and Pippin) were free.
