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In this joint work with late Pawel Domanski, we give certain conditions on linear dynamical properties of the weighted backward shift operators $B_w : A(\Omega) \to A(\Omega),$ with weight sequences $w = (w_n)_{n\in \mathbb{N}}$, acting on the spaces of real analytic functions $A(\Omega)$ on open subsets $\Omega$ of $\mathbb{R}$ containing zero, that send the monomials $x^n$ to $w_nx^{n-1}$ for all $n\in \mathbb{N}$ and the unit function to the zero function. A useful tool for these characterizations is the description of the point spectra and the eigenspaces of weighted backward shifts on $A(\Omega)$.