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For $ 1 \\leq j \\leq d $, denote the $ j $-th the Lyapunov exponent by $ \\chi_{j} := \\sum_{i\\in\\Lambda} - p_{i} \\log | r_{i,j} |$, and define the IFS induced by $ \\Phi $ on the $j$-th coordinate as $ \\Phi_{j} := \\{ x \\mapsto r_{i,j}x + t_{i,j}\\}_{i\\in\\Lambda}$. 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