Chaotic dynamical systems are difficult to simulate because of their strong nonlinearity, long-term aperiodic behavior, and sensitivity to initial conditions, especially in turbulent-flow applications where fine-grid simulations are computationally costly and reduced-order models can introduce significant error. This work investigates deep generative learning, particularly generative adversarial networks (GAN), as a synthetic modeling approach for invariant state snapshots of chaotic systems. By linking ergodic theory with generative learning, it establishes that GAN training converges mathematically for ergodic systems and that statistics computed from GAN-generated snapshots converge to simulation-based statistics. The approach is demonstrated on systems ranging from the Lorenz attractor and Kuramoto–Sivashinsky equation to turbulent flows around a cylinder and under periodic wake impact, where deep convolutional GAN reproduce chaotic and turbulent snapshots accurately, efficiently, and without prescribed physical boundary conditions.