Editing Dyadic rational (section)

===Functions with dyadic rationals as distinguished points===
{{multiple image|total_width=480
|image1=Minkowski question mark.svg|caption1=[[Minkowski's question-mark function]] maps rational numbers to dyadic rationals|alt1=Graph of the question mark function
|image2=Daubechies4-functions.svg|caption2=A [[Daubechies wavelet]], showing points of non-smoothness at dyadic rationals|alt2=Graph of the scaling and wavelet functions of Daubechies' wavelet
}}
Because they are a dense subset of the real numbers, the dyadic rationals, with their numeric ordering, form a [[dense order]]. As with any two unbounded countable dense linear orders, by [[Cantor's isomorphism theorem]],{{r|bmmn}} the dyadic rationals are [[Order isomorphism|order-isomorphic]] to the rational numbers.  In this case, [[Minkowski's question-mark function]] provides an order-preserving [[bijection]] between the set of all rational numbers and the set of dyadic rationals.{{r|girgensohn}}

The dyadic rationals play a key role in the analysis of [[Daubechies wavelet]]s, as the set of points where the [[Wavelet#Scaling function|scaling function]] of these wavelets is non-smooth.{{r|pollen}} Similarly, the dyadic rationals parameterize the discontinuities in the boundary between stable and unstable points in the parameter space of the [[Hénon map]].{{r|cvi-gun-pro}}

The set of [[piecewise linear function|piecewise linear]] [[homeomorphism]]s from the [[unit interval]] to itself that have power-of-2 slopes and dyadic-rational breakpoints forms a group under the operation of [[function composition]]. This is [[Thompson groups|Thompson's group]], the first known example of an infinite but [[Presentation of a group|finitely presented]] [[simple group]].{{r|brin}} The same group can also be represented by an action on rooted binary trees,{{r|can-flo}} or by an action on the dyadic rationals within the unit interval.{{r|cor-guy-pit}}