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The problem of the existence of an additional (independent on the energy) first integral, of a geodesic (or magnetic geodesic) flow, which is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial…
The squarefree flow is a natural dynamical system whose topological and ergodic properties are closely linked to the behavior of squarefree numbers. We prove that the squarefree flow carries a unique measure of maximal entropy and express…
Continuing the work in \cite{ergodic-infinite}, we show that within each stratum of translation surfaces, there is a residual set of surfaces for which the geodesic flow in almost every direction is ergodic for almost-every periodic group…
The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at…
This paper answers some questions about conjugate points along the geodesics corresponding to steady 2D Euler flows, posed by a paper of Drivas-Misiolek-Shi-Yoneda. We present a new sufficient criterion for the existence of conjugate…
We report and analyze the results of numerical studies of dense granular flows in two and three dimensions, using both linear damped springs and Hertzian force laws between particles. Chute flow generically produces a constant density…
For ideal fluid flow with zero surface tension and gravity, it remains unknown whether local singularities on the free surface can develop in well-posed initial value problems with smooth initial data. This is so despite great advances over…
It is proved that a certain type of monotone flow has a global period provided periodic points are dense.
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points.…
We study the subsequential convergence of singular solutions to the Ricci flow with prescribed constant in space geodesic curvature on compact surfaces with boundary. Furthermore, we show that in the particular case of rotational symmetry,…
We study steady-state properties of inelastic gases in two-dimensions in the presence of an energy source. We generalize previous hydrodynamic treatments to situations where high and low density regions coexist. The theoretical predictions…
In this work, we study ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior, the main feature is a condition of transversality between unstable leaves when projected by the stable…
On geometrically finite negatively curved surfaces, we give necessary and sufficient conditions for a one-sided horocycle $(h^s u)_{s\ge 0}$ to be dense in the nonwandering set of the geodesic flow. We prove that all dense one-sided orbits…
We study suspension flows defined over sub-shifts of finite type with continuous roof functions. We prove the existence of suspension flows with uncountably many ergodic measures of maximal entropy. More generally, we prove that any…
The equations for geodesic flow on the ellipsoid are well known, and were first solved by Jacobi in 1838 by separating the variables of the Hamilton-Jacobi equation. In 1979 Moser investigated the case of the general ellipsoid with distinct…
In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by $-c^2$ is of Anosov type, then the constant of contraction of the flow is $\geq e^{-c}$. Moreover,…
We investigate fundamental nonlinear dynamics of ferrofluidic Taylor-Couette flow - flow confined between two concentric independently rotating cylinders - consider small aspect ratio by solving the ferrohydrodynamical equations, carrying…
We consider a class of inhomogeneous self-similar cosmological models in which the perfect fluid flow is tangential to the orbits of a three-parameter similarity group. We restrict the similarity group to possess both an Abelian $G_{2}$,…
We establish the existence and uniqueness of some smooth accelerating transonic flows governed by the three dimensional steady compressible Euler equations with an external force in cylinders with arbitrary cross sections, which include…
The geodesic structure is very closely related to the trace of the Laplace operator, involved in the calculation of the expectation value of the energy momentum tensor in Universes with non trivial topology. The purpose of this work is to…