Participation in permissionless blockchains results in competition over system resources, which needs to be controlled with fees. Until recently, Ethereum's fee mechanism was implemented via a first-price auction that resulted in unpredictable fees as well as other inefficiencies. Launched on August 5, 2021, EIP-1559 is an improved proposal that introduces a number of innovative features such as a dynamically adaptive basefee that is burnt, instead of being paid to the miners. Despite intense interest in understanding its properties, several basic questions such as whether and under what conditions does this protocol self-stabilize have remained elusive thus far.

We perform a thorough analysis of the resulting fee market dynamic mechanism via a combination of tools from game theory as well as dynamical systems. We start by providing bounds on the step-size of the base-fee update rule that suffice for global convergence to equilibrium via Lyapunov arguments. In the negative direction, we show that for larger step-sizes instability and even formal Li-Yorke chaos are possible under a wide range of settings. We complement these topological results with quantitative bounds on the possible range of basefees. We conclude our analysis with a thorough experimental case study that corroborates our theoretical findings.