Tire tracks geometry, hatchet planimeter, Menzin's conjecture and oscillation of unicycle tracks

The model of a bicycle is a unit segment AB that can move in the plane so that it remains tangent to the trajectory of point A (the rear wheel, fixed on the bicycle frame); the same model describes the hatchet planimeter. I shall discuss two tire track problems. The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This mapping of a circle to a circle is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. Moebius transformations belong to one of the three types: elliptic, parabolic and hyperbolic. I shall outline a proof of a 100 years old conjecture: if the front wheel track is an oval with area at least Pi then the respective monodromy is hyperbolic. I shall also discuss bicycle motions such that the track of the rear wheel coincides with that of the front wheel. I shall explain why such ''unicycle" tracks become more and more oscillating in forward direction and cannot be infinitely extended backward.