We study the trace map for the period doubling substitution, proving its orbits are forward bounded, a sufficient condition for application in the study of spectral measures. This result is in contrast with some recent findings of unbounded orbits on the trace map for the Thue–Morse substitution.

中文翻译：

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周期图加倍替换的跟踪图的前向有界性

我们研究了周期图加倍替换的轨迹图，证明了它的轨道是前向有界的，这是在光谱测量研究中应用的充分条件。这一结果与最近在Thue-Morse替代轨迹图上发现的无界轨道相反。