Related papers: On the volume of singular-hyperbolic sets

We relate dominated splitting for a linear multiplicative cocyle with dominated splitting for the exterior powers of this cocycle. For a C1 vector field X on a 3-manifold, we can obtain singular-hyperbolicity using only the tangent map DX…

The coexistence of singularities and regular orbits in chain transitive sets has been a major obstacle in understanding the hyperbolic/partial hyperbolic nature of robust dynamics. Notably, the vector fields with all periodic orbits…

We call a partially hyperbolic diffeomorphism \emph{partially volume expanding} if the Jacobian restricted to any hyperplane that contains the unstable bundle $E^u$ is larger than $1$. This is a $C^1$ open property. We show that any…

We study the finiteness of uniform sinks for flow. Precisely, we prove that, for $\alpha>0$ $T>0$, if a vector field $X$ has only hyperbolic singularities or sectionally dissipative singularities, then $X$ can have only finitely many…

In this paper, we consider the contracting curvature flow of smooth closed surfaces in $3$-dimensional hyperbolic space and in $3$-dimensional sphere. In the hyperbolic case, we show that if the initial surface $M_0$ has positive scalar…

This paper proves the following: A volume preserving vector field on a compact 3-manifold whose dual 2-form is exact can not generate uniquely ergodic dynamics unless its asymptotic linking number is zero.

In this paper, we mainly study immersed self-expander hypersurfaces in Euclidean space whose mean curvatures have some linear growth controls. We discuss the volume growths and the finiteness of the weighted volumes. We prove some theorems…

This paper belongs to a series of papers devoted to the study of the structure of the non-wandering set of an A-diffeomorphism. We study such set $NW(f)$ for an $\Omega$-stable diffeomorphism $f$, given on a closed connected 3-manifold…

In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in…

We consider an autonomous system of partial differential equations for one-dimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases…

We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non-hyperbolic fillings;…

On every compact 3-manifold, we build a non-empty open set $\cU$ of $\Diff^1(M)$ such that, for every $r\geq 1$, every $C^r$-generic diffeomorphism $f\in\cU\cap \Diff^r(M)$ has no topological attractors. On higher dimensional manifolds, one…

We consider the class of partially hyperbolic diffeomorphisms $f:M\to M$ obtained as the discretization of topological Anosov flows. We show uniqueness of minimal unstable lamination for these systems provided that the underlying Anosov…

This study focuses on the impact of the cosmological constant on hyperbolically symmetric matter configurations in static background. We extend the work of Herrera \cite{herrera2021hyperbolically} and describe the influences of such…

In this expository paper, we review the history and the recent breakthroughs in the spectral theory of large volume hyperbolic surfaces. More precisely, we focus mostly on the investigation of the first non-trivial eigenvalue $\lambda_1$…

Let W be a compact manifold and let \rho be a representation of its fundamental group into PSL(2,C). The volume of \rho is defined by taking any \rho-equivariant map from the universal cover of W to H^3 and then by integrating the pull-back…

In this paper, we study the existence of SRB measures and their properties for infinite dimensional dynamical systems in a Hilbert space. We show several results including (i) if the system has a partially hyperbolic attractor with…

We give a new proof of the well-known result that the minimal volume vector fields on $\mathbb{S}^3(r)$ are the Hopf vector fields. Such proof relies again on calibration theory, arising here from a systematic point of view given by a…

We prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C2-robustly on the metric. Similar results are obtained when considering even feeble properties like the weak shadowing and the…

We present numerical verification of hyperbolic nature for chaotic attractor in a system of two coupled non-autonomous van der Pol oscillators (Kuznetsov, Phys. Rev. Lett., 95, 144101, 2005). At certain parameter values, in the…