A split leaf:
The growth of the Lulav leaf: The leaves of the Lulav grow in a pair of two leaves which are parallel to each other. The two leaves are attached to each other by their back and open in their front.
If the leaves of the Lulav split: If majority of the leaves of the Lulav have separated in the majority of their length, the Lulav is invalid.
If the leaf grew without being a double leaf: If the majority of the leaves grew without being double paired on majority of their length the Lulav is invalid. The same applies if majority of the leaves grew with separated pairs it is invalid. [If however the leaves were double sided in majority of their length the Lulav is valid.]
The law of the Tiyomes:
What is the Tiyomes? Special attention must be given to the middle upper leaf of the Lulav called the Tiyomes. The Tiyomes is the middle upper most leaf which extends from the spine. It grows in the center of the Lulav and is considered the head of the Lulav. [If a Lulav has two middle leaves of equal height some Poskim rule that both leaves have the status of a Tiyomes.]
The law if the leaf split entirely: If the Tiyomes grew without a double sided leaf the Lulav is invalid even if all the other leaves are double sided. Likewise if the Tiyomes grew double sided and afterwards split entirely from the top until the spine, it is invalid.
The law if the leaf partially split: Lechatchilah it is a Mitzvah Min Hamuvchar to buy a Lulav whose double leafed Tiyomes is completely attached to each other (at their back) from its top to the point the other leaves begin shooting out from the spine. If this is not available, or it split after buying it, then the Lulav remains valid so long as the leaf is not completely separated from top to bottom [where the other leaves begin growing from it]. Furthermore even if one has a friend which owns a Lulav with a complete Tiyomes one is not required to use it on the first day of Sukkos and rather may use his Lulav that has a partially split Tiyomes.
On Chol Hamoed: Has the following leniencies. A Hemnick split is kosher. The Tiyomes is kosher even if completely split.
On the second day of Sukkos in Chutz Laaretz: If no other Lulav is available he may use it, but without a blessing.
Must the Tiyomes be double leafed throughout its entire length?
It is implied from Admur that it is a Mitzvah Min Hamuvchar for the Tiyomes to be entirely double leafed from the top of the Tiyomes until the area of the spine, and so rule some Rabbanim. However others learn that it is not necessary for the Tiyomes to be entirely double leafed and so is the ruling of many great Rabbanim.
What if the Tiyomes is split only at its tip as is common to occur?
It is a Mitzvah Min Hamuvchar to purchase a Tiyomes that is completely closed including its tip.
If the split of the Tiyomes is only noticeable after close examination is it initially invalid?
Some Poskim rule that initially one is to purchase a Lulav that its Tiyomes is fully closed even after proper contemplation. [It is not necessary to use a magnifying glass.]
Does it help to glue a split Tiyomes leaf back together?
Some Poskim rule it is valid to glue the Tiyomes together.
 As the Lulav is not considered Hadar. [ibid]
 M”B 645/15
The reason: The invalidation of this middle leaf, the Tiyomes, is due to incompletion, as the Torah states “Ulikachtem Lachem” and the Sages expounded that it must be complete and whole . Now this middle leaf is considered the centerpiece of the Lulav as one sees it at his first sight and hence if it is completely split this matter is readily apparent and the Lulav is not considered complete. [ibid]
 The reason: The reason for this is because some opinions rule that if the Tiyomes is even partially split the Lulav is invalid. Now although we do not rule like this opinion nevertheless it is proper to suspect for this opinion when possible. [ibid]
According to the Alter Rebbe this is defined as a Hemnik split [a split in the Lulav which makes it appear as two] of which the Gemara states that the Lulav is invalid. This invalidation only applies to the spine of the Lulav and not to its Tiyomes. Other Poskim however rule that a Hemnick split is defined as a fork like appearance on the top of the Lulav by the Tiyomes. [See Kashrus Daled Minim p. 83-84; Piskeiy Teshuvos 645/6 and 10-11]
 The reason: The reason for this is because the Lulav appears lacking and incomplete. [ibid]
 Kashrus Daled Minim [Shtern] in name of Rav Elyashiv based on Admur 645/9
 Rav SZ”A and others mentioned in Piskeiy Teshuvos 645/5 footnote 24
 Piskeiy Teshuvos 645/9
 Piskeiy Teshuvos 645/9; Rav Akiva Eiger Shut
 Piskeiy Teshuvos 645/4