Hausdorff and packing dimensions and measures for nonlinear transversally non-conformal thin solenoids

2020/03/19 by Mohammadpour, Reza, Przytycki, Feliks, Rams, Michal
#37D25 #37D35 #Dynamical Systems (math.DS) #FOS: Mathematics #Primary: 37D20. Secondary: 28A78

paper · doi:10.48550/arxiv.2003.08926

Abstract

We extend results by B. Hasselblatt, J. Schmeling in Dimension product structure of hyperbolic sets (2004), and by the third author and K. Simon in Hausdorff and packing measures for solenoids (2003), for C^1+ε hyperbolic, (partially) linear solenoids Λ over the circle embedded in ℝ³ non-conformally attracting in the stable discs W^s direction, to nonlinear ones. Under an assumption of transversality and assumptions on Lyapunov exponents for an appropriate Gibbs measure imposing thinness, assuming also there is an invariant C^1+ε strong stable foliation, we prove that Hausdorff dimension \rm HD(Λ∩ W^s) is the same quantity t₀ for all W^s and else \rm HD(Λ)=t₀+1. We prove also that for the packing measure 0

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