Volume 29, Number 3

In this article, we consider mechanical billiard systems defined with Lagrange's integrable {extension} of Euler's two-center problems in the Euclidean space, the sphere, and the hyperbolic space of arbitrary dimension $n \geqslant 3$. In the three-dimensional Euclidean space, we show that the billiard systems with any finite combinations of {spheroids and circular hyperboloids of two sheets} having two foci at the Kepler centers are integrable. The same holds for the projections of these systems on the three-dimensional sphere and in the three-dimensional hyperbolic space by means of central projection. Using the same approach, we also extend these results to the $n$-dimensional cases.

We construct $C^{\infty}$ time-periodic fluctuating surfaces in $\mathbb R^3$ such that the corresponding non-autonomous geodesic flow has orbits along which the energy, and thus the speed goes to infinity. We begin with a static surface $M$ in $\mathbb R^3$ on which the geodesic flow (with respect to the induced metric from $\mathbb R^3$) has a hyperbolic periodic orbit with a transverse homoclinic orbit. Taking this hyperbolic periodic orbit in an interval of energy levels gives us a normally hyperbolic invariant manifold $\Lambda$, the stable and unstable manifolds of which have a transverse homoclinic intersection. The surface $M$ is embedded into $\mathbb R^3$ via a near-identity time-periodic embedding $G: M \to \mathbb R^3$. Then the pullback under $G$ of the induced metric on $G(M)$ is a time-periodic metric on $M$, and the corresponding geodesic flow has a normally hyperbolic invariant manifold close to $\Lambda$, with stable and unstable manifolds intersecting transversely along a homoclinic channel. Perturbative techniques are used to calculate the scattering map and construct pseudo-orbits that move up along the cylinder. The energy tends to infinity along such pseudo-orbits. Finally, existing shadowing methods are applied to establish the existence of actual orbits of the non-autonomous geodesic flow shadowing these pseudo-orbits. In the same way we prove the existence of oscillatory trajectories, along which the limit inferior of the energy is finite, but the limit superior is infinite.

In this paper, we consider the equation $u_{xx}+Q(x)u+P(x)u^3=0$ where $Q(x)$ and $P(x)$ are periodic functions. It is known that, if $P(x)$ changes sign, a ``great part'' of the solutions for this equation are singular, i.e., they tend to infinity at a finite point of the real axis. Our aim is to describe as completely as possible solutions, which are regular (i.e., not singular) on $\mathbb{R}$. For this purpose we consider the Poincaré map $\mathcal{P}$ (i.e., the map-over-period) for this equation and analyse the areas of the plane $(u,u_x)$ where $\mathcal{P}$ and $\mathcal{P}^{-1}$ are defined. We give sufficient conditions for hyperbolic dynamics generated by $\mathcal{P}$ in these areas and show that the regular solutions correspond to a Cantor set situated in these areas. We also present a numerical algorithm for verifying these sufficient conditions at the level of ``numerical evidence''. This allows us to describe regular solutions of this equation, completely or within some class, by means of symbolic dynamics. We show that regular solutions can be coded by bi-infinite sequences of symbols of some alphabet, completely or within some class. Examples of the application of this technique are given.

It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs.
Estimating the exact values of both the metric mean dimension and mean Hausdorff dimension for a homeomorphism is a challenging task. We need to establish a precise relationship between the sizes of the horseshoes and the number of appropriated legs to control both quantities.
Let $N$ be an $n$-dimensional compact Riemannian manifold, where $n \geqslant 2$, and $\alpha \in [0, n]$. In this paper, we construct a homeomorphism $\phi: N \rightarrow N$ with mean Hausdorff dimension equal to $\alpha$. Furthermore, we prove that the set of homeomorphisms on $N$ with both lower and upper mean Hausdorff dimensions equal to $\alpha$ is dense in $\text{Hom}(N)$. Additionally, we establish that the set of homeomorphisms with upper mean Hausdorff dimension equal to $n$ contains a residual subset of $\text{Hom}(N).$

We prove here the criterion of $C^1$- $\Omega$-stability of self-maps of a 3D-torus, which are skew products of circle maps. The $C^1$- $\Omega$-stability property is studied with respect to homeomorphisms of skew products type. We give here an example of the $\Omega$-stable map on a 3D-torus and investigate approximating properties of maps under consideration.