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arXiv · 2501.15940

Equivalent Conditions for Domination of $\mathrm{M}(2,\mathbb{C})$-sequences

Chang Sun·Zhenghe Zhang
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Abstract

It is well known that a $\mathrm{SL}(2,\mathbb{C})$-sequence is uniformly hyperbolic if and only it satisfies a uniform exponential growth condition. Similarly, for $\mathrm{GL}(2,\mathbb{C})$-sequences whose determinants are uniformly bounded away from zero, it has dominated splitting if and only if it satisfies a uniform exponential gap condition between the two singular values. Inspired by [QTZ], we provide a similar equivalent description in terms of singular values for $\mathrm{M}(2,\mathbb{C})$-sequences that admit dominated splitting. We also prove a version of the Avalanche Principle for such sequences.

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Chang Sun, Zhenghe Zhang. 2025-01-27. Equivalent Conditions for Domination of $\mathrm{M}(2,\mathbb{C})$-sequences. https://arxiv.org/abs/2501.15940

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