The aim of this paper is to provide alternative characterizations of hyperbolic affine generalized infinite iterated function systems. More precisely, we prove that, for such a system $${\mathcal {F}}=((X,\left\| .\right\| ),(f_{i})_{i\in I})$$ F = ( ( X , . ) , ( f i ) i ∈ I ) , among others, the following statements are equivalent: (a) $${\mathcal {F}}$$ F is hyperbolic. (b) $$ {\mathcal {F}}$$ F has attractor. (c) $${\mathcal {F}}$$ F is strictly topologically contractive. (d) $${\mathcal {F}}$$ F is uniformly point-fibred. In this way we generalize the result from the paper by Miculescu and Mihail (J Math Anal Appl 407:56–68, 2013). More equivalent statements are given for the particular case when I is finite and X is finite dimensional.

中文翻译：

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关于双曲仿射广义无限迭代函数系统

本文的目的是提供双曲仿射广义无限迭代函数系统的替代表征。更准确地说，我们证明，对于这样的系统 $${\mathcal {F}}=((X,\left\| .\right\| ),(f_{i})_{i\in I}) $$ F = ( ( X , . ) , ( fi ) i ∈ I ) ，其中，以下陈述是等价的： (a) $${\mathcal {F}}$$ F 是双曲线的。(b) $$ {\mathcal {F}}$$ F 有吸引子。(c) $${\mathcal {F}}$$ F 是严格拓扑收缩的。(d) $${\mathcal {F}}$$ F 是均匀点纤维。通过这种方式，我们概括了 Miculescu 和 Mihail 的论文 (J Math Anal Appl 407:56–68, 2013) 的结果。当 I 是有限的且 X 是有限维的时，针对特定情况给出了更多等价的陈述。