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We give a new sufficient condition for the Julia set of a real analytic function which is a periodic point of renormalization to have Hausdorff dimension less than 2. This condition can be verified numerically. We present results of computer experiments suggesting that this condition is satisfied for real periodic points of renormalization with low periods. Our results support the conjecture that all real Feigenbaum maps have Julia sets of Hausdorff dimension less than 2. 2020 Mathematicsdoi:10.4064/ba210406-10-4 fatcat:om7d55li3bep3gfsn4vcxb64nu