Anisotropic 2D FUP and quantum open baker's map

We prove an essential spectral gap for 2D anisotropic quantum open baker’s map. This extends the 1D results of Dyatlov–Jin 2017 and the isotropic 2D results of Cohen 2025a. The key ingredient is the anisotropic discrete fractal uncertainty principle (FUP) associated with a 2D anisotropic fractal set called the Bedford–McMullen carpet. We also study the relation between our anisotropic discrete FUP and its continuous counterpart in the spirit of Dyatlov–Jin 2018 and Cohen 2025a. In particular, we prove {continuous FUP} for 2D {anisotropic porous} sets, extending the (high-dimensional) isotropic results of Cohen 2025b. To the best of our knowledge, the anisotropic (line) porosity condition – a variant of Cohen’s line porosity and stronger than ball porosity – appears to be new to the literature.