Oak stair landing joints splitting
Due to climate changes in your house (moisture and heat fluctuations) and the natural tendency for wood to be effected from them, wood will move. The drier and more unsealed the wood is the more it will swell and contract. Also depending on how it was milled and what section of the tree it was cut from will determine how much moisture it can absorb and release.   You are correct in surmising that the landing shouldn't be splitting. Although wood can't be prevented from moving altogether there are steps that can account for and limit this movement. When constructing a project made from wood the planks  moisture level should be around 15%. If they are to used indoors with a central heating system an even lower level of 11% should be used. A kiln is used to get this final moisture level.  If the landing was installed using strip (1 1/2-2 inch wide) flooring that was milled with a tongue and groove the tongue should be nailed to the sub floor and into the floor joists if possible. If the boards are wider than 4-6 inches they should be installed in such a way that the fastener keeps them flat but can also allow for slight shifting. This is sometimes accomplished by oval screw holes from underneath.   If the board split is slight epoxy filler/adhesive can be color matched to the wood species to make the patch less noticeable. I would be hesitant to use any other type of filler for a floor patch since the nature of the wood is to move the patch will eventually get pushed loose.