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

• Published in: Ergodic theory and dynamical systems 2022-11, Vol.42 (11), p.3458-3489
• Main Authors: MOHAMMADPOUR, REZA, PRZYTYCKI, FELIKS, RAMS, MICHAŁ
• Format: Article
• Language: English
• Subjects: Approximation; Liapunov exponents; Original Article; Solenoids; Theorems
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2021.94

We extend the results of Hasselblatt and Schmeling [Dimension product structure of hyperbolic sets. Modern Dynamical Systems and Applications. Eds. B. Hasselblatt, M. Brin and Y. Pesin. Cambridge University Press, New York, 2004, pp. 331–345] and of Rams and Simon [Hausdorff and packing measure for solenoids. Ergod. Th. & Dynam. Sys. 23 (2003), 273–292] for $C^{1+\varepsilon }$ hyperbolic, (partially) linear solenoids $\Lambda $ over the circle embedded in $\mathbb {R}^3$ non-conformally attracting in the stable discs $W^s$ direction, to nonlinear solenoids. Under the assumptions of transversality and on the Lyapunov exponents for an appropriate Gibbs measure imposing thinness, as well as the assumption that there is an invariant $C^{1+\varepsilon }$ strong stable foliation, we prove that Hausdorff dimension $\operatorname {\mathrm {HD}}(\Lambda \cap W^s)$ is the same quantity $t_0$ for all $W^s$ and else $\mathrm {HD}(\Lambda )=t_0+1$ . We prove also that for the packing measure, $0