The flat spot standard family: variation of the entrance time median

Abstract

We completely determine the median of the entrance time function for the one-parameter standard family of flat spot maps. The function mapping parameters to medians is shown to be a countably piecewise affine degree-one homeomorphism, with uncountably many cusps and precisely nine critical points.

Notes

Notes

1. Throughout we shall use ∫ g to denote the Lebesgue integral of a function g on the whole circle 𝕋, and ∫ _A  g or for its Lebesgue integral over the set A or interval [a, b].

2. The terminology Sturmian is due to Morse and Hedlund 15, who studied the 0–1 sequences corresponding to the binary expansions of points in Sturmian sets. These Sturmian sequences have since then been extensively investigated, see e.g. 5,14,16–19 for further characterizations and properties.

3. Uniqueness is because e _γ is strictly positive on 𝕋 ∖ F _γ = [γ, γ + 1/2].