^{1,a)}, Franz Gähler

^{1,b)}and Uwe Grimm

^{2,c)}

### Abstract

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is known as the Devil's staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures. For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta functions. This is achieved as a corollary of analysing the Anderson-Putnam complex for the determination of the cohomological invariants of the corresponding tiling spaces.

We are grateful to Michel Dekking, Xinghua Deng, Natalie Frank, and Daniel Lenz for helpful discussions. This work was supported by the German Research Council (Deutsche Forschungsgemeinschaft (DFG)), within the CRC 701, and by a Leverhulme Visiting Professorship grant (M.B.). We thank the Erwin Schrödinger Institute in Vienna for hospitality, where part of the work was done.

I. INTRODUCTION II. A SUMMARY OF THE CLASSIC THUE-MORSE SEQUENCE III. A FAMILY OF GENERALISED THUE-MORSE SEQUENCES IV. THE DIFFRACTION MEASURE OF THE GTM SYSTEM V. TOPOLOGICAL INVARIANTS VI. DYNAMICAL ZETA FUNCTIONS VII. FURTHER DIRECTIONS

### Key Topics

- Cumulative distribution functions
- 18.0
- Eigenvalues
- 17.0
- Dirac equation
- 10.0
- Functional equations
- 7.0
- Topology
- 7.0

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### Abstract

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is known as the Devil's staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures. For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta functions. This is achieved as a corollary of analysing the Anderson-Putnam complex for the determination of the cohomological invariants of the corresponding tiling spaces.

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