How Do Pool Fees Affect the Long-Term Growth or Decline of the $k$ Constant?

Pool fees, which are a small percentage of each trade, are typically reinvested back into the pool's reserves, increasing the quantities of $x$ and $y$ and thus increasing the constant $k$. Over the long term, if trading volume is consistently high, the compounding effect of these fees will lead to a sustained growth in $k$.

This growth offsets impermanent loss and causes the LP tokens to appreciate in value over time.