Quasi-multiplicativity of Typical Cocycles

Abstract

We show that typical (in the sense of [BV04] and [AV07]) Hölder and fiber-bunched GL d (R)-valued cocycles over a subshift of finite type are uniformly quasimultiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding (necessarily unique) equilibrium state for such cocycles, and apply this result to repellers. Moreover, we show that the pointwise Lyapunov spectrum is closed and convex, and establish partial multifractal analysis on the level sets of pointwise Lyapunov exponents for such cocycles.

Thermodynamic formalism of \text{GL}_2(\mathbb{R}) -cocycles with canonical holonomies

“…[16] proves Proposition 5 in the setting of fiber-bunched GL d (R)cocycles. As noted in Remark 7, weak typicality in this paper defined as in Definition 1.3 is more general and differs slightly from typicality defined in [3] and [16]. However, Proposition 5 still holds for weakly typical cocycles with little modifications, and we briefly sketch the proof in the following subsection.…”

“…Moreover, the bounded distortion property (5) of the norm potentials for cocycles in H allows the result to be extended to norm potentials of quasimultiplicative cocycles. Such result for fiber-bunching cocycles C α b (Σ T , GL d (R)) are established in [16]. The only use of fiber-bunching assumption there is to establish the convergence as well as the Hölder continuity of the canonical holonomies H s/u , and hence, the same result holds for A ∈ H as well.…”

“…The following proposition from the second author's previous result [16] states that weakly typical coycles are quasi-multiplicative.…”

“…We now sketch the proof of Proposition 5 by following the proof of [16,Theorem A]. The proof outlined below is simpler than the original proof appearing in [16] as we are working with GL 2 (R)-cocycles.…”

“…We now sketch the proof of Proposition 5 by following the proof of [16,Theorem A]. The proof outlined below is simpler than the original proof appearing in [16] as we are working with GL 2 (R)-cocycles. For statements not fully elaborated in the sketch of the proof, we refer the readers to [16,Section 4] for details.…”