We utilize a chartist-fundamentalist model to examine the limits of informationally efficient stock markets. In this model, chartists are permanently active in the stock market, while fundamentalists trade only when their mispricing-dependent trading signals are sufficiently strong. As a result, the model dynamics are driven by a two-dimensional piecewise-linear discontinuous map. Our findings suggest the possible coexistence of two distinct regimes. Depending on the initial conditions, the stock market may exhibit either constant or oscillatory mispricing. Constant mispricing occurs when chartists remain the sole active speculators, causing the stock price to converge toward a nonfundamental value. Conversely, the stock price oscillates around its fundamental value when fundamentalists repeatedly enter and exit the market. Interestingly, these oscillatory dynamics are associated with a new type of attractor, termed a “weird quasiperiodic attractor”. When subjected to dynamic noise, our model reproduces several important stylized facts of stock markets and can thus be considered validated.