Abstract
AbstractWe 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<\Pi _{t_0}(\Lambda \cap W^s)<\infty $ , but for Hausdorff measure, $\mathrm {HM}_{t_0}(\Lambda \cap W^s)=0$ for all $W^s$ . Also $0<\Pi _{1+t_0}(\Lambda ) <\infty $ and $\mathrm {HM}_{1+t_0}(\Lambda )=0$ . A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every $W^s$ has measure $\mathrm {HM}_{t_0}$ equal to 0 and even Hausdorff dimension less than $t_0$ . The latter holds due to a large deviations phenomenon.
Highlights
We extend the results of Hasselblatt and Schmeling [Dimension product structure of hyperbolic sets
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 C1+ε strong stable foliation, we prove that Hausdorff dimension HD( ∩ W s) is the same quantity t0 for all W s and else HD( ) = t0 + 1
We prove that for the packing measure, 0 < t0 ( ∩ W s) < ∞, but for Hausdorff measure, HMt0 ( ∩ W s ) = 0 for all W s
Summary
Under the assumptions of Theorem 1.2, for t denoting packing measure in dimension t, for every x ∈ S1, it holds 0 < t0 ( ∩ Wxs) < ∞. This follows (and clarifies) [9]. We prove that a bigger set has measure 0, the set of p which are not strong Lipschitz, called above weak non-Lipschitz For such a p, the projection πx,y(W u(p)) intersects some πx,y(W u(q)) values for q ∈/ W u(p) arbitrarily close to W u(p).
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