Stack-sorting with consecutive-pattern-avoiding stacks

We introduce consecutive-pattern-avoiding stack-sorting maps SC?, which are natural generalizations of West's map s and analogues the classical-pattern-avoiding s? recently introduced by Cerbai, Claesson, Ferrari. characterize patterns ? such that Sort(SC?), set permutations sortable via s?SC?, is a permutation class, we enumerate sets Sort(SC?) for ??{123,132,321}. also study SC? from dynamical point view, characterizing periodic points all ??S3 computing max??Sn?|SC??1(?)| ??{132,213,231,312}. In addition, s132, show maximum number iterations s132 needed to send in Sn n?1. The paper ends with numerous open problems conjectures.