Unique equilibrium states, large deviations and Lyapunov spectra for the Katok Map

We study the thermodynamic formalism of a $C^{\mathrm{\infty }}$ non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a H"older continuous potential with one additional condition, or the geometric t-potential ${\phi }_t$ with $t<1$, the equilibrium state exists and is unique. We derive the level-2 large deviation principle for the equilibrium state of ${\phi }_t$. We study the multifractal spectra of the Katok map for the entropy and dimension of level sets of Lyapunov exponents.