The Lap-Counting Function for Linear Mod One and Tent Maps
Dr. Leopold Flatto (Bell Laboratories)
The study of interval maps is a basic topic in dynamical systems, these
maps arising in diverse settings such as population genetics and number
theory. A central problem in the subject is to decide when two such maps
are topologically conjugate. An important invariant is the lap-counting
function $L(z)=\sum L_n z^n$, where $L_n$ is the number of monatonic
pieces of the $n$th iterate of the map. This function was introduced by
Milnor and Thurston, who used it to show that interval maps are
semi-conjugate to piecewise linear maps with slopes $\pm s$, where $s$ is
the topological entropy.
The linear mod one and test maps are the simplest examples of such maps.
For these we obtain a complete description of $L(z)$ are related to the
topological dynamical and ergodic properties of the maps.
Finally, linear mod one maps are related to the dynamic of the Lorenz
attractor, discovered by Lorenz in his study of weather prediction.