Anisotropic 2D FUP and quantum open baker's map

Abstract

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.