Consider a polynomial (of any degree) with an attracting periodic point. We give a combinatorial description of its Julia set near the boundary of its hyperbolic component. Moreover, assuming that there is a unique boundary critical point (of any order), we can prove local connectivity at the boundary if the critical combinatorics is not strongly recurrent. Lastly, we characterize the strong recurrence property in terms of a new kind of renormalization acting on the Bernoulli shift.