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.