Related papers: Absolutely singular dynamical foliations

Transverse one dimensional foliations play an important role in the study of codimension one foliations. In \cite{KR2}, the authors introduced the notion of flow box decomposition of a 3-manifold $M$. This is a decomposition of $M$ that…

We show that certain types of the three-legged accessibility property of a partially hyperbolic diffeomorphism imply the existence of a unique minimal set for one strong foliation and the transitivity of the other one. In case the center…

We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic.

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and…

We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological entropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.

For a pair of bounded linear Hilbert space operators $A$ and $B$ one considers the Lebesgue type decompositions of $B$ with respect to $A$ into an almost dominated part and a singular part, analogous to the Lebesgue decomposition for a pair…

We prove that the Oseledets splittings of an ergodic hyperbolic measure of a $C^{1+ r}$ diffeomorphism can be approximated by that of atomic measures on hyperbolic periodic orbits. This removes the assumption on simple spectrum in…

Stationary measures on the circle that arise from a large class of random walks on the fundamental group of a finite-area complete hyperbolic surface with cusps are singular with respect to the Lebesgue measure. In particular, it is…

We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semi-conjugate to nonuniformly expanding maps. Using the theory of projective metric on cones we then prove…

Metric entropies along a hierarchy of unstable foliations are investigated for $C^1$ diffeomorphisms with dominated splitting. The analogues of Ruelle's inequality and Pesin's formula, which relate the metric entropy and Lyapunov exponents…

We prove that a transversely holomorphic foliation which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not of zero measure. Similarly, we prove that a finitely generated subgroup of…

We consider the open set constructed by M. Shub in [42] of partially hyperbolic skew products on the space $\mathbb{T}^2\times \mathbb{T}^2$ whose non-wandering set is not stable. We show that there exists an open set $\mathcal{U}$ of such…

This paper gives a complete classification of the possible ergodic decompositions for certain open families of volume-preserving partially hyperbolic diffeomorphisms. These families include systems with compact center leaves and…

In this paper we mainly deal with an invariant (ergodic) hyperbolic measure $\mu$ for a diffeomorphism $f,$ assuming that $f$ is just $C^1$ and for $\mu$ a.e. $x$, the sum of Oseledec spaces corresponding to negative Lyapunov exponents…

We prove that the restriction of a probability measure invariant under a nonhyperbolic, ergodic and totally irreducible automorphism of a compact connected abelian group to the leaves of the central foliation is severely restricted. We also…

We study the stability of pullback foliations under morphisms and rational maps via Grothendieck's Drapeaux scheme. In the local setting, a foliated version of Schlessinger's Theorem on rigidity of conical singularities was achieved. We…

We show that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism then it also admits an Anosov flow. Moreover, we give a complete classification of partially hyperbolic diffeomorphism in hyperbolic 3-manifolds as well as…

We study the topological properties of the leaves of the singular foliation induced by a closed 1-form of Morse type on a compact orbifold. In particular, we establish criteria that characterize when all such leaves are compact, when they…

We give a sketch of proof that any two (Lebesgue) measurable subsets of the unit sphere in $R^n$, for $n\ge 3$, with non-empty interiors and of the same measure are equidecomposable using pieces that are measurable.

We show that the open unit ball $\mathbb{B}^n$ of $\mathbb{C}^n$ $(n>1)$ admits a nonsingular holomorphic foliation by complete properly embedded holomorphic discs.