For each primitive Dirichlet character , a hypothesis is formulated in terms of zeros of the associated -function . It is shown that for any such character, is equivalent to the Generalized Riemann Hypothesis.
In rotations with a binary symbolic dynamics, a critical curve is the locus of parameters for which the boundaries of the partition that defines the symbolic dynamics are connected via a prescribed number of iterations and symbolic itinerary. We study the arithmetical and geometrical properties of these curves in parameter space.
In this article we define the semigroup associated to a primitive substitution. We use it to construct a minimal automaton which generates a substitution sequence in reverse reading. We show, in the case where the substitution has a coincidence, that this automaton completely describes the semicocycle discontinuities of .
We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.
We show that any union of finitely many shifted model sets from a given cut-and-project scheme is a model set in some modified cut-and-project scheme. Restricting to direct space , we show that any inter-model set is a model set in some modified cut-and-project scheme with second countable internal space. In both cases, the window in the modified cut-and-project scheme inherits the topological and measure-theoretic properties of the original windows.
A flow view is the graph of a measurable conjugacy between a substitution or S-adic subshift and an exchange of infinitely many intervals in , where is Lebesgue measure. A natural refining sequence of partitions of is transferred to using a canonical addressing scheme, a fixed dual substitution , and a shift-invariant probability measure . On the flow view, is shown horizontally at a height of using colored unit intervals to represent the letters.
The infinite interval exchange transformation is well approximated by exchanges of finitely many intervals, making numeric and graphic methods possible. We prove that in certain cases a choice of dual substitution guarantees that is self-similar. We discuss why the spectral type of is of particular interest. As an example of utility, some spectral results for constant-length substitutions are included.
The pair correlations of the Thue–Morse sequence and system are revisited, with focus on asymptotic results on various means. First, it is shown that all higher-order correlations of the Thue–Morse sequence with general real weights are effectively determined by a single value of the balanced 2-point correlation. As a consequence, we show that all odd-order correlations of the balanced Thue–Morse sequence vanish, and that, for any even , the -point correlations of the balanced Thue–Morse sequence have mean value zero, as do their absolute values, raised to an arbitrary positive power. All these results also apply to the entire Thue–Morse system. We finish by showing how the correlations of the Thue–Morse system with general real weights can be derived from the balanced 2-point correlations.
In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit a growth behaviour which is impossible for substitutions on finite alphabets. While for both settings the leading term of the tile counting function is exponential (and guided by the inflation factor), the behaviour of the second-order term is strikingly different. For the finite setting, it is known that the second term is also exponential or exponential times a polynomial. We exhibit a large family of examples where the second term is at least exponential in divided by half-integer powers of , where is the number of substitution steps. In particular, we provide an identity for this discrepancy in terms of linear combinations of Catalan numbers.
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