stable foliation
Recently Published Documents


TOTAL DOCUMENTS

21
(FIVE YEARS 2)

H-INDEX

5
(FIVE YEARS 1)

2021 ◽  
pp. 1-32
Author(s):  
REZA MOHAMMADPOUR ◽  
FELIKS PRZYTYCKI ◽  
MICHAŁ RAMS

Abstract 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<\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.


2018 ◽  
Vol 39 (12) ◽  
pp. 3169-3184 ◽  
Author(s):  
WAEL BAHSOUN ◽  
MARKS RUZIBOEV

We prove statistical stability for a family of Lorenz attractors with a $C^{1+\unicode[STIX]{x1D6FC}}$ stable foliation.


2017 ◽  
Vol 39 (1) ◽  
pp. 74-104 ◽  
Author(s):  
RICARDO T. BORTOLOTTI

In this work, we analyze ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior; the main feature is a condition of transversality between the projections of unstable leaves, projecting through the stable foliation. We prove that partial hyperbolic attractors satisfying this condition of transversality, neutrality in the central direction and regularity of the stable foliation admit a finite number of physical measures, coinciding with the ergodic u-Gibbs States, whose union of the basins has full Lebesgue measure. Moreover, we describe the construction of robustly non-hyperbolic attractors satisfying these properties.


2011 ◽  
Vol 21 (01) ◽  
pp. 255-273 ◽  
Author(s):  
SHULIANG SHUI ◽  
JINGJING LI ◽  
XUYANG ZHANG

Heteroclinic bifurcations in four-dimensional vector fields are investigated by setting up local coordinates near a heteroclinic loop. This heteroclinic loop consists of two principal heteroclinic orbits, but there is one stable foliation that involves an inclination flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit, and 1-periodic orbit are studied. Also, the nonexistence, existence of the 2-homoclinic and 2-periodic orbit are demonstrated.


2002 ◽  
Vol 1 (4) ◽  
pp. 513-541 ◽  
Author(s):  
Christian Bonatti ◽  
Lorenzo J. Díaz ◽  
Raúl Ures

We give a topological criterion for the minimality of the strong unstable (or stable) foliation of robustly transitive partially hyperbolic diffeomorphisms.As a consequence we prove that, for $3$-manifolds, there is an open and dense subset of robustly transitive diffeomorphisms (far from homoclinic tangencies) such that either the strong stable or the strong unstable foliation is robustly minimal.We also give a topological condition (existence of a central periodic compact leaf) guaranteeing (for an open and dense subset) the simultaneous minimality of the two strong foliations.AMS 2000 Mathematics subject classification: Primary 37D25; 37C70; 37C20; 37C29


Sign in / Sign up

Export Citation Format

Share Document