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Given a self-similar iterated function system $\\Phi=(E, \\{f_i\\}_{i=1}^m)\\in\\mathcal E$ on the real line, for each point $x\\in E$ we can find a sequence $(i_k)=i_1i_2\\ldots\\in\\{1,\\ldots,m\\}^\\mathbb N$, called a coding of $x$, such that $$ x=\\lim_{n\\to\\infty}f_{i_1}\\circ f_{i_{2}}\\circ\\cdots\\circ f_{i_n}(0). $$ For $k=1,2,\\ldots, \\aleph_0$ or $2^{\\aleph_0}$ we investigate the subset $\\mathcal U_k(\\Phi)$ which consists of all $x\\in E$ having precisely $k$ different codings. 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Given a self-similar iterated function system $\\Phi=(E, \\{f_i\\}_{i=1}^m)\\in\\mathcal E$ on the real line, for each point $x\\in E$ we can find a sequence $(i_k)=i_1i_2\\ldots\\in\\{1,\\ldots,m\\}^\\mathbb N$, called a coding of $x$, such that $$ x=\\lim_{n\\to\\infty}f_{i_1}\\circ f_{i_{2}}\\circ\\cdots\\circ f_{i_n}(0). $$ For $k=1,2,\\ldots, \\aleph_0$ or $2^{\\aleph_0}$ we investigate the subset $\\mathcal U_k(\\Phi)$ which consists of all $x\\in E$ having precisely $k$ different codings. 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