  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 were
introduced and investigated in [Y. Saiki, H. Takahasi, J. A. Yorke,
Nonlinearity, 34 (2021), 5744--5761] as minimal models of the unstable
dimension variability in multidimensional dynamical systems. We show that
natural coding spaces of these maps coincide with the Dyck system that has come
from the theory of languages. Based on this coincidence, we start to develop a
complementary analysis on their invariant measures. As a first attempt, we show
the existence of two ergodic measures of maximal entropy for the generalized
heterochaos baker maps. We also clarify a mechanism for the breakdown of
entropy approachability.
