In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of self-affine sets, and the theory of random matrix products, it has often been found useful to assume positivity of the matrix entries in order to simplify or make feasible certain types of calculation. It is natural to ask how positivity may be relaxed or generalised in a way which enables similar calculations to be made in more general contexts. On the one hand one may generalise by considering almost additive or asymptotically additive potentials which mimic the properties enjoyed by the logarithm of the norm of a positive matrix cocycle; on the other hand one may consider matrix cocycles which are dominated, a condition which includes positive matrix cocycles but is more general. In this article we explore the relationship between almost additivity and domination for planar cocycles. We show in particular that a locally constant linear cocycle in the plane is almost additive if and only if it is either conjugate to a cocycle of isometries, or satisfies a property slightly weaker than domination which is introduced in this paper. Applications to matrix thermodynamic formalism are presented.

中文翻译：

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平面矩阵共环的支配、几乎可加性和热力学形式

在诸如线性共环的热力学形式主义、自仿射集的维数理论和随机矩阵乘积理论等主题中，经常发现假设矩阵项的正性以简化或使某些类型可行是有用的的计算。很自然地会问，如何以一种能够在更一般的上下文中进行类似计算的方式来放松或概括积极性。一方面，可以通过考虑几乎可加或渐近可加的势来进行概括，这些势模拟了正矩阵共环范数的对数所享有的性质；另一方面，可以考虑占主导地位的矩阵共环，这种情况包括正矩阵共环但更一般。在本文中，我们探讨了平面共环的几乎可加性和支配性之间的关系。我们特别表明，当且仅当它与等距的共环共轭，或者满足略弱于本文中介绍的支配的性质时，平面中的局部常数线性共环几乎是可加的。介绍了矩阵热力学形式主义的应用。