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We consider the stochastic dynamics of the pure and random ferromagneticIsing model on the hierarchical diamond lattice of branching ratio $K$ withfractal dimension $d_f=(\ln (2K))/\ln 2$. We adapt the Real SpaceRenormalization procedure introduced in our previous work [C. Monthus and T.Garel, J. Stat. Mech. P02037 (2013)] to study the equilibrium time $t_{eq}(L)$as a function of the system size $L$ near zero-temperature. For the pure Isingmodel, we obtain the behavior $t_{eq}(L) \sim L^{\alpha} e^{\beta 2J L^{d_s}} $where $d_s=d_f-1$ is the interface dimension, and we compute the prefactorexponent $\alpha$. For the random ferromagnetic Ising model, we derive therenormalization rules for dynamical barriers $B_{eq}(L) \equiv (\lnt_{eq}/\beta)$ near zero temperature. For the fractal dimension $d_f=2$, weobtain that the dynamical barrier scales as $ B_{eq}(L)= c L+L^{1/2} u$ where$u$ is a Gaussian random variable of non-zero-mean. While the non-random termscaling as $L$ corresponds to the energy-cost of the creation of a system-sizedomain-wall, the fluctuation part scaling as $L^{1/2}$ characterizes thebarriers for the motion of the system-size domain-wall after its creation. Thisscaling corresponds to the dynamical exponent $\psi=1/2$, in agreement with theconjecture $\psi=d_s/2$ proposed in [C. Monthus and T. Garel, J. Phys. A 41,115002 (2008)]. In particular, it is clearly different from the dropletexponent $\theta \simeq 0.299$ involved in the statics of the randomferromagnet on the same lattice.

Dynamical barriers of pure and random ferromagnetic Ising models on fractal lattices