Non-existence of Lyapunov exponents in the Newhouse domain

We show that within the Newhouse domain of $C^r$ surface diffeomorphisms ($r\in [2,\mathrm{\infty })$), there exists a dense subset $\mathcal{D}$ such that for any $f\in \mathcal{D}$, Lyapunov exponents fail to exist for all points in some open set $U$ and all nonzero tangent vectors in some open cone $V\subset {\mathbb{R}}^2$. This demonstrates that the non-existence of Lyapunov exponents is a persistent phenomenon in the setting of robust homoclinic tangencies. The proof relies on constructing diffeomorphisms exhibiting specific oscillatory return times near a homoclinic tangency, incorporating techniques from Newhouse theory and recent results on Lyapunov irregularity, alongside several refinements and new arguments.