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Moreover it follows from Theorem 2.5 that these vector fields are orthogonal geodesic vector fields, i.e integral curves are geodesics. If the boundary of a billiard desk Q(A)????????Q(A) doesn't intersect zone 3, then the assertion immediately follows from the construction of the structural multilayered elliptic flowers and from the optical definition of ellipses. It is simple to see that the boundary of a structural elliptic flower built over sq. comprises two pairs of arcs of confocal ellipses (Fig. 1). Therefore, on this case there are not less than two elliptic periodic orbits (of period two) in the core. The same is true for the 4 pairs of the arcs of confocal ellipses in case when A????A is a sq.. It is plausible that equation (1) is in fact irrelevant in increased dimensions, i.e. S????S have to be a sphere additionally within the case 2. of Theorem 1.2. However we were not able to prove this. This argument implies that every one factors of S????S are umbilical and therefore S????S is a spherical sphere. Therefore divergence of the orbits beats convergence at any link of an orbit which connects two points of different boundary parts.

We say that two billiards (or polygons) are order equal if each of the billiards has an orbit whose footpoints are dense within the boundary and the two sequences of footpoints of those orbits have the identical combinatorial order. Can order equivalent polygons have different (variety of) angles, or at the least different lengths of sides? Then a particular one layer elliptic flower is a billiard table with n????n boundary parts, which are the arcs of ellipses with the focuses located at the ends of the small diagonals of A????A and the ends on two semi-strains orthogonal to the (intersecting) sides of A????A, which the corresponding small diagonal connects (Fig. 4). If A????A is a triangle, then the sides play also the position of small diagonals. Let now the boundary of Q(A)????????Q(A) intersects zone 3 (see Fig.2(b)). The projection of the core Co(M(A))????????????????Co(M(A)) to the billiard table Q(A)????????Q(A) can be known as a base core and denoted by Co(Q(A))????????????????Co(Q(A)). Remark 3. Observe that the chaotic cores of the special one layer elliptic flowers are nonempty subsets of the bottom polygon A????A. Intriguing could possibly be also the long run studies of quantum elliptic flowers billiards. Therefore the primary natural common question for the longer term studies of elliptic flowers is to grasp how the modifications within the dynamics of these billiards happen.

One might count on that the research of elliptic flowers with the higher ranges petals will deliver some new examples of an unseen before behaviors, which are amenable to rigorous studies. There are nonetheless, at the least theoretical, potentialities that there exist elliptic flowers billiards, which have some fascinating properties not thought-about in the current paper. Another interesting concern, though probably basically a technical one, is about building of elliptic flowers with some specific properties for any convex polygon taken as a base. We give now the best example of an elliptic flower with all the properties considered within the theorems 3.7, 3.8 and 3.9. (It can be done by choosing b????b either large or small enough). Take sufficiently giant radius R????R for an ergodic circular flower over A????A. A????A. In what follows we are going to refer to such diagonals of any polygon as to small diagonals (simply having in mind that in a common convex polygon not all small diagonals may very well be also the shortest ones). The centers of all ellipses are on the centers of the corresponding shortest (small) diagonals of the essential polygon A????A.

Substitute now every of those n????n circles by the tangent to them ellipses with the focuses situated at the centers of the corresponding shortest diagonals of A????A and the small axis orthogonal to these diagonals. We are able to keep this property for an elliptic flower by selecting sufficiently massive (with respect to the scale of the base polygon A????A) circles. Exactly the same proof, as the certainly one of Theorem 3.8 goes once more for special one layer elliptic flowers with not less than 5 sided common base polygon by perturbing a hyperbolic and ergodic circular flower with giant circles. However, all orbits of the core can't have two consecutive collisions with one and the identical regular component of the boundary. Therefore each of the circles containing a part of this circular flower intersects only two neighboring boundary parts. There exist hyperbolic elliptic flowers billiards with section area consisting of three ergodic parts of constructive measure, that are two chaotic tracks and a chaotic core. Denote by M(A)????????M(A), Q(A)????????Q(A) the phase house and the configuration area (billiard table) constructed over the base polygon A????A.