Non-linear iterations and higher splitting

We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $κ$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $κ$. Moreover assuming GCH, $κ^{<κ}=κ$, we show that: (1) if $κ$ is strongly unfoldable, $κ^+\leqβ=\hbox{cf}(β)\leq \hbox{cf}(δ)\leqδ\leqμ$ and $\hbox{cf}(μ)>κ$,then there is a cardinal preserving generic extension in which $\mathfrak{s}(κ)=κ^+\leq\mathfrak{b}(κ)=β\leq\mathfrak{d}(κ)=δ\leq 2^κ=μ.$ (2) if $κ$ is strongly inaccessible, $λ>κ^+$, then in the generic extension obtained as the $<κ$-support iteration of $κ$-Hechler forcing of length $λ$ there are no $κ$-towers of length $λ$.

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