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We present the numerical flow iteration (NuFI) for solving the Vlasov--Poisson equation. In a certain sense specified later herein, NuFI provides infinite resolution of the distribution function. NuFI exactly preserves positivity, all $L^p$-norms, charge, and entropy. Numerical experiments show no energy drift. NuFI is fast, requires several orders of magnitude less memory than conventional approaches, and can very efficiently be parallelised on GPU clusters. Low fidelity simulations provide good qualitative results for extended periods of time and can be computed on low-cost workstations.