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We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from the diffeomorphism group of the sphere to the orthogonal group. This result…
We establish a stable homotopy-theoretic version of a recent result of Farber and Weinberger on the fibrewise topological complexity of sphere bundles and prove, by closely parallel methods, a similar result for real, complex and…
We discuss recent progress in understanding the dynamical properties of partially hyperbolic diffeomorphisms that preserve volume. The main topics addressed are density of stable ergodicity and stable accessibility, center Lyapunov…
A long-standing conjecture asserts that any Anosov diffeomorphism of a closed manifold is finitely covered by a diffeomorphism which is topologically conjugate to a hyperbolic automorphism of a nilpotent manifold. In this paper, we show…
We provide abstract conditions which imply the existence of a robustly invariant neighbourhood of a global section of a fibre bundle flow. We then apply such a result to the bundle flow generated by an Anosov flow when the fibre is the…
We consider a hyperbolic toral automorphism $L$ and its $C^1$-small perturbation $f$. It is well-known that $f$ is Anosov and topologically conjugate to $L$, but a conjugacy $H$ is only H\"older continuous in general. We discuss conditions…
We show that every partially hyperbolic diffeomorphism with a 1-dimensional center bundle has a principal symbolic extension. On the other hand, we show there are no symbolic extensions $C^1$-generically among diffeomorphisms containing…
We prove a conjecture of J. Palis: any diffeomorphism of a compact manifold can be C1-approximated by a Morse-Smale diffeomorphism or by a diffeomorphism having a transverse homoclinic intersection. ----- Cr'eation d'intersection homoclines…
We prove that every smooth diffeomorphism group valued cocycle over certain abelian Anosov actions on tori (and more generally on infranilmanifolds), is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at…
We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic…
We prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially…
We slightly extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. Spin structures are natural…
We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and -- for almost all primes $p$ -- $\mathbb{F}_p$-homology growth above the…
We show that in the absence of periodic centre annuli, a partially hyperbolic surface endomorphism is dynamically coherent and leaf conjugate to its linearisation. We proceed to characterise the dynamics in the presence of periodic centre…
According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this article is whether these type of…
Let ($M$, $\Omega$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^l$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose…
We consider principal fibre bundles with a given connection and construct almost complex structures on the total space if the adjoint bundle is isomorphic to the tangent bundle of the base. We derive the integrability condition. If the…
We present a general construction of embedded minimal and constant mean curvature surfaces in $\mathbb{S}^n$ and one-phase free boundaries joined by a smooth interpolation by capillary hypersurfaces. This framework recovers all known…
In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to…
Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply…