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We introduce two parametrized families of piecewise affine maps on $[0,1]^2$and $[0,1]^3$, as generalizations of the heterochaos baker maps which wereintroduced and investigated in [Y. Saiki, H. Takahasi, J. A. Yorke,Nonlinearity, 34 (2021), 5744--5761] as minimal models of the unstabledimension variability in multidimensional dynamical systems. We show thatnatural coding spaces of these maps coincide with the Dyck system that has comefrom the theory of languages. Based on this coincidence, we start to develop acomplementary analysis on their invariant measures. As a first attempt, we showthe existence of two ergodic measures of maximal entropy for the generalizedheterochaos baker maps. We also clarify a mechanism for the breakdown ofentropy approachability.

Heterochaos baker maps and the Dyck system: maximal entropy measures and a mechanism for the breakdown of entropy approachability