On the smallest numerical semigroups closed under affine maps

We study numerical semigroups $S_{a,b}(m)$ generated by the orbit of $m$ under the affine map $T_{a,b}(z)=az+b$, where $a\ge 2$, $m>1$, $\gcd(b,m)=1$, and $b\ge -(a-2)m-2$. This extends the usual affine-closed setting to feasible negative values of $b$. We write $A_i=(a^i-1)/(a-1)$ and let $n$ be the smallest positive integer such that $A_n\ge m$. We determine the minimal generators and give an explicit description of the Ap\'ery set, obtaining homogeneity and formulas for the Frobenius number and the genus. We also study pseudo-Frobenius numbers via the induced Ap\'ery parametrization and prove the sharp upper bound $\operatorname{t}(S_{a,b}(m))\le n-1$ for the type. We give a complete characterization of the symmetric members of the family in terms of the canonical representative of $m-1$. Finally, we exhibit a subfamily whose pseudo-Frobenius numbers form an arithmetic progression of length $n-1$; in particular, this subfamily attains the bound.