Livsic theorems for connected Lie groups

Abstract

Abstract. Let φ be a hyperbolic diffeomorphism on a basic set Λ and let G be a connected Lie group. Let f : Λ → G be Hölder. Assuming that f satisfies a natural partial hyperbolicity assumption, we show that if u : Λ → G is a measurable solution to f = uφ · u −1 a.e., then u must in fact be Hölder. Under an additional centre bunching condition on f , we show that if f assigns 'weight' equal to the identity to each periodic orbit of φ, then f = uφ · u −1 for some Hölder u. These results extend well-known theorems due to Livšic when G is compact or abelian.

Livšic Theorem for matrix cocycles

“…the highlights are [5], [2], [3], [12], [13], [14], [15], [16], [17], [19]. We refer the reader to [4] and to the upcoming book [9] for some of the most recent results and overview of historical development in this area.…”

Smooth cocycle rigidity for expanding maps, and an application to Mostow rigidity

“…More recent contributions to Livsic theory have focused on extending rigidity results to cocycles with values in non-compact groups (see in particular [13,23,28]). Many cocycle rigidity results have analogues in the context of hyperbolic flows (see in particular [30]).…”

A unified framework for Schelling’s model of segregation

“…There is variation in the nature of the equilibrium and the mechanism of relaxation. Some models (such as that of Laurie and Jaggi [29]) arrive at a stable configuration which does not change, either because every agent is satisfied or because no acceptable move exists; for certain models these limit states have been studied mathematically [23,24]. Other models, particularly those which allow random moves that lower the satisfaction of the agent involved (for example Gauvin et al [18]) reach something resembling a thermal equilibrium composed of many similar states.…”

“…Aside from a few mathematical papers investigating limit states of deterministic versions of the model [23,24], analytical results on Schelling-like models are conspicuous in their absence. Surprisingly, this situation persists even in the physics literature where the typical models proposed are still too complicated to admit a successful theoretical treatment and must instead be simulated.…”