Related papers: A dynamical construction of Liouville domains
In this paper we study transitivity of partially hyperbolic endomorphisms of the two torus whose action in the first homology has two integer eigenvalues of moduli greater than one. We prove that if the Jacobian is everywhere greater than…
In hyperbolic dynamics, a well-known result is: every hyperbolic Lyapunov stable set, is attracting; it's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics. This question is still open, although some…
For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristics semiflows associated with certain Hamilton-Jacobi equations, and build the relation between the $\omega$-limit set of this…
We generalized several results for the arithmetic dynamics of monomial maps, including Silverman's conjectures on height growth, dynamical Mordell-Lang conjecture, and dynamical Manin-Mumford conjecture. These results are originally known…
We introduce a new class of Sturm-Liouville operators with periodically modulated parameters. Their spectral properties depend on the monodromy matrix of the underlying periodic problem computed for the spectral parameter equal to $0$.…
Parallel to the study of toric domains, symplectically convex, and dynamically convex domains in $(\mathbb R^4, \omega_{\rm std})$, we build an analogous framework and corresponding subclasses for Liouville domains in $(T^*\mathbb…
We consider the St\"ackel transform, also known as the coupling-constant metamorphosis, which under certain conditions turns a Hamiltonian dynamical system into another such system and preserves the Liouville integrability. We show that the…
A method of integrable discretization of the Liouville type nonlinear partial differential equations is suggested based on integrals. New examples of discrete Liouville type models are presented.
We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians $H$ obtained as one-dimensional extensions of natural (geodesic) $n$-dimensional Hamiltonians $L$. The…
We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold-Liouville theorem: the system not need to be integrable on the whole phase space, while the invariant hypersurface is…
We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their…
We present some constructions that are merely the fruit of combining recent results from two areas of smooth dynamics: nonuniformly hyperbolic systems and elliptic constructions.
Given a strictly concave rational PL function $\phi$ on a complete $n$-dimensional fan $\Sigma$, we construct an exact symplectic structure of finite volume on $(\mathbb{C}^\times)^n$ and a family of functions $H_{\phi,\epsilon}$ called…
A method of defining the complex structure(moduli) for dynamically triangulated(DT) surfaces with torus topology is proposed. Distribution of the moduli parameter is measured numerically and compared with the Liouville theory for the…
We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms.
We consider Liouville-type and partial regularity results for the nonlinear fourth-order problem $$ \Delta^2 u=|u|^{p-1}u\ \{in} \ \R^n,$$ where $ p>1$ and $n\ge1$. We give a complete classification of stable and finite Morse index…
We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic…
We consider a discrete dynamical system on a pseudo-Riemannian manifold and we determine the concept of a hyperbolic set for it. We insert a condition in the definition of a hyperbolic set which implies to the unique decomposition of a part…
We construct a transient bounded-degree graph no transient subgraph of which embeds in any surface of finite genus. Moreover, we construct a transient, Liouville, bounded-degree, Gromov--hyperbolic graph with trivial hyperbolic boundary…
The low-energy physics of (quasi)degenerate one-dimensional systems is typically understood as the particle-like dynamics of kinks between stable, ordered structures. Such dynamics, we show, becomes highly non-trivial when the ground states…