Recognisability for generalised hierarchical pattern spaces of finite local complexity

Abstract

We develop a general framework of Euclidean patterns and pattern spaces of translational finite local complexity (FLC), analogues of translational tiling spaces. The notion of a self affine substitution of tilings is extended to both individual patterns and pattern spaces, which we ask are mapped onto by a local derivation map (the analogue of a sliding block code map from Symbolic Dynamics) from its own expansion by a linear map. We prove recognisability for these, with no minimality requirements. In particular we show that, for each pattern in the space, its substitutional pre-images are translation equivalent. Sizes of all fibres are then determined by relative groups of translational periods. This answers an open question of Cortez and Solomyak, on whether non-periodic tilings necessarily have unique pre-images under substitution: they do, even for a wider notion of pattern and being substitutional. It is shown that there exists a power of the substitution under which any given pattern of the pattern space has multiple pre-images if and only if it has disconnected group of periods. This is equivalent to being periodic in the standard return discrete cases of tilings and Delone sets, but our results also cover examples with non-discrete groups of periods, such as spaces of uniformly discrete but non-relatively dense point sets.