Orbits in Teichmüller dynamics admits a critical exponent gap

Abstract

McMullen '03 constructs a collection of orbits ${\mathrm{SL}}_2(\mathbb{R}).x$ in $\mathcal{H}(1,1)$ with infinitely generated stabilizers ${\mathrm{stab}}_{{\mathrm{SL}}_2(\mathbb{R})}(x)$. We prove a gap in the set of critical exponents of stabilizers of ${\mathrm{SL}}_2(\mathbb{R})$-orbits in ${\mathcal{H}}_g$: for every $x\in {\mathcal{H}}_g$, either ${\mathrm{stab}}_{{\mathrm{SL}}_2(\mathbb{R})}(x)$ is a lattice, or we have a uniform bound on the critical exponent $\hat{I}´({\mathrm{stab}}_{{\mathrm{SL}}_2(\mathbb{R})}(x))\le 1-{\epsilon }_g$.

21 pages