Bitcoin is represented as a digital ledger. It does not "split" like you are thinking. It is divisible. One Bitcoin can be divided into 100 million smaller pieces called satoshis. Therefore, if I had 1BTC and sent someone half of it, I am sending them 50 million satoshis. Ultimately, it is still just digits on a digital ledger. There is nothing physical about it, so there is neither "splitting" nor "joining".
But when you make multiple small transactions over time, does that clutter your wallet with too many inputs, or does it affect the transaction fees later on? Just trying to understand how it all adds up in the background.
If for example you have multiple inputs coming from fractions of bitcoin sent to your wallet over time, they are called UTXOs and if you want to make a transaction that you've to use two or more of those inputs, it increases the transaction weight as compared to making the transaction from just one UTXO and you would end up paying more fees. A way to go around this is targeting when the fee rate is low and consolidating your multiple UTXOs to a single address (more like joining them together as a single UTXO in a new address still in the same wallet if you wish). This is achieved by sending all of your inputs to a new address and paying a little fee since the fee rate is low as against possibly initiating a transaction when the fee rate is high and paying higher fees each time you make a transaction that involves more that a single UTXO in your wallet.