So is Avalokiteśvara reborn into more than one body and if so is every being that is Avalokiteśvara the same in some way? Or is that just different Tibetan traditions? Generally is rebirth restricted to one body or can it be split across more than one? I appreciate that this will be different across branches of Buddhism so can I restrict my question to Tibetan Buddhism.
It's closer to the Hindu idea of an avatar. For the majority of people, not particularly accomplished, rebirth functions as normal. Those others aren't being "reborn" because the source of their form-body hasn't died.
Because Buddhism denies independent existence of substantial "self", it does not posit literal reincarnation, as in someone literally being a come back of e.g. Ananda (Buddha's favorite student).
Tibetan Buddhism does have a notion of Tulku though, which is similar. Tulku is a new person that inherits the life stories and responsibilities of a previous person. This is done through teaching this person to deliberately identify with their predecessor, to learn predecessor's history, and to carry on his or her work.
Because the new person's life is so heavily influenced and inspired by the predecessor, we can say that the new person is an embodiment of the abstract energy of the previous one.
In this sense, multiple people could embody the same predecessor. In fact, it is more often than not a case with Great teachers, that they have multiple "emanations" (usually, of mind, of speech, and of enlightened activity) -- because a regular limited person can't give justice to all of that greatness, so declaring one Tulku would not be appropriate.
As Andrei has said the Tulku system can give rise to multiple rebirths. The maximum number of rebirths is five which correspond to the following manifestations
- guna (quality)
However even in the event of a multiple rebirth, one of the manifestations will be seen as the primary one. In Bernardo Bertolucci film the Little Buddha a Tibetan lama is reborn into 3 Western children. The rebirths are separate manifestations of his body, speech and mind.
In the interest of correct referencing, this information comes from the book Exploring Karma and Rebirth by Nagapriya.
There most definitely are reincarnations of specific beings in Buddhism. HH the Dalai Lama may have the emanation qualities of the Buddhist Avaloketeshvara, who was one of the disciples of Shakyamuni Buddha, but he is a direct reincarnation of all the prior Dalai Lamas. So are the Karmapa Rinpoche's, the Tai Situ Rinpoches, and Namkhai Nyingpo Rinpoche's. Most of what are identified as Tulkus in Tibetan Buddhism are emanations of a human being that attained enlightenment, and then emanated specific Buddha Family Qualities (body, speech, mind, activity, qualities) into an accomplished disciple, or other being very accomplished yet not enlightened, that the Enlightened Khandro or Lama believed could carry out the Buddha Activities of the particular emanation of the Enlightened Mind bestowed upon them (the emanation) by the gone beyond Enlightened Being/Lama/Khandro.
The five qualities are not rebirth or reincarnation qualities; they are qualities of the emanation rebirth. It is not a reincarnation rebirth.
There are five Khyentse emanations of Jamyang Kyentse Wangp. There are five emanation rebirths of Jamyang Khyentse Chokyi Lodro (Dzongsar Khyentse Rinpoche is only one emanation of Jyamgyan Khyentse Chokye Lodro, He is Not, I repeat, Not, the reincarnation of Jamyang Khyentse Chokyo Lodro, but is the reincarnation of some other (to me) unidentified person who was blessed with one of the 5 emanation qualities of Jamyang Khyentse Chokyi Lodro. There is no "maximum number of rebirths", nor is there a maximum number of emanations, although the five qualities listed previously by me are a more typical type of emanation activity by an Enlightened Being. However, the latter can emanate endlessly, as they are truly enlightened. Is there a Buddhist Studies Program at a college near where you are living? Or can the participants of this blog obtain access to the library of such a program, or visit one? Seems there is a lot of confusion out there, and spreading via the internet is not such a good idea. There is a wonderful resource online: treasuryoflives.org. Also, it is necessary to study with a living Buddhist Master, to clarify questions such as the ones raised by this blog. I challenge you all to question my response by responsibly engaging in intelligent productive research, via legitimate libraries of Buddhist Studies Programs at the graduate university level, and via qualified Tibetan Buddhist Teachers. Good luck!