There has been enormous interest in these maps, as they arose from #integrable systems, like the Sine-Gordon equation. The interest grew with the construction of discrete #Painlevé equations. The autonomous cases of such equations are QRT maps.

`In #mathematics, the Lebesgue differentiation #theorem is a theorem of real #analysis, which states that for almost every point, the value of an #integrable #function is the limit of #infinitesimal averages taken about the point. The theorem is named for Henri #Lebesgue. `

https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem

Lebesgue differentiation theorem - Wikipedia