We study the small scale structure of axion dark matter in the\npost-inflationary scenario, which predicts the formation of low-mass,\nhigh-density clumps of gravitationally bound axions called axion miniclusters.\nTo this end, we follow numerically the cosmological evolution of the axion\nfield and the network of strings and domain walls until the density contrast is\nfrozen. Our simulations, comprising up to $8192^3$ points, are the largest\nstudies of the axion field evolution in the non-linear regime presented so far.\nAxitons, pseudo-breathers of the Klein-Gordon equation, are observed to form in\nour simulation at late times. Studying their properties analytically and\nnumerically, we observe that in particular, the earliest axitons contribute to\ndensity perturbations at the typical length scale of miniclusters. We analyse\nthe small scale structure of the density field, giving the correlation length,\npower spectrum and the distribution of high-density regions that will collapse\ninto axion miniclusters. The final density field of our simulations can be used\nto calculate the minicluster mass fraction in simulations including gravity. In\nparticular, we find that typical minicluster progenitors are smaller than\npreviously thought and only of moderate, $\\mathcal{O}(1)$ overdensity. We\nexpect these miniclusters to have a rich sub-structure, emerging from\nsmall-scale fluctuations produced in the collapse of the string-wall network\nand from axitons.\n