Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure

Abstract

Given a non-conformal repeller Λ of a C 1+γ map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok's approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure. 0

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

“…An A-invariant sub-bundle F ⊂ E is α-Hölder if and only if it is H s/u -invariant.Remark 6. While [2, Corollary 3.5] is stated for fiber-bunched cocycles, the same result holds for α-Hölder cocycles whose canonical holonomies converge and are α-Hölder continuous, including α-Hölder cocycles satisfying(8). Moreover, Proposition 3 and 4 readily extend to our setting where the base dynamical system is a mixing subshift of finite type (Σ T , σ).…”

“…This implies the continuity of the pressure P(Φ A ) and the equilibrium state µ A restricted to the set of weakly typical cocycles U w ; see [16,Theorem B]. In this direction of results, we remark that Cao, Pesin, and Zhao [8] recently established a more general result that the map A → P(Φ A ) is continuous on C α (Σ T , GL d (R)) using a different approach. See also [12].…”

Quasi-multiplicativity of Typical Cocycles

“…Cao, Pesin, and Zhao [CPZ18] recently proved a result that implies Theorem B (1). See Section 3 for further comments.…”

Subadditive Pre-Image Variational Principle for Bundle Random Dynamical Systems

“…The subadditive topological pressure has been proven to be essential to estimate the lower bound of the Hausdorff dimension of non-conformal repellers in the deterministic dynamical systems [32]. Thus, it should be asked if there exists a random version of pre-image thermodynamic formalism for subadditive sequences of potentials, which might have some potential applications in the study of multifractal analysis of non-conformal random dynamical systems.…”