A noncircular oval with convex unit-tangent iterates

For an oriented regular plane curve $\gamma$, define $\mathcal{T}\gamma =\gamma +\tau$, where $\tau$ is its unit tangent. We construct a noncircular curve ${\mathrm{\Gamma }}_0$ such that every iterate ${\mathcal{T}}^n{\mathrm{\Gamma }}_0$, $n\ge 0$, is a smooth embedded closed curve with everywhere positive curvature. This answers a question of Tabachnikov. The construction starts with a noncompact convex curve $C$, asymptotic at its two ends to parallel lines, satisfying $\mathcal{T}C=C+(V,0)$ for some $V>0$. From $C$ we construct long closed convex curves. After matching perimeters, we compare each curve with a convex curve whose unit-tangent image is the next curve in the sequence. In suitable arclength coordinates, their curvature functions differ by an exponentially small amount in $L^1$. Uniform inverse estimates turn these approximate relations into an exact sequence ${\mathrm{\Gamma }}_{n+1}=\mathcal{T}{\mathrm{\Gamma }}_n$. Each ${\mathrm{\Gamma }}_n$ lies in a strip of width bounded independently of $n$, whereas the radii of iterated circles tend to infinity.