Topology automaton and Hölder equivalence of Barański carpets

The study of Lipschitz equivalence of fractals is a very active topic in recent years. In 2023, Huang \emph{et al.} ($\mathit{\text{Topology}}\mathit{\text{automaton}}\mathit{\text{of}}\mathit{\text{self-similar}}\mathit{\text{sets}}\mathit{\text{and}}\mathit{\text{its}}\mathit{\text{applications}}\mathit{\text{to}}\mathit{\text{metrical}}\mathit{\text{classifications}}$, Nonlinearity $\mathbf{\text{36}}$ (2023), 2541-2566.) studied the Hölder and Lipschitz equivalence of a class of p.c.f. self-similar sets which are not totally disconnected. The main tool they used is the so called topology automaton. In this paper, we define topology automaton for Barański carpets, and we show that the method used in Huang \emph{et al.} still works for the self-affine and non-p.c.f. settings. As an application, we obtain a very general sufficient condition for Barański carpets to be Hölder (or Lipschitz) equivalent.