Abstract:ABSTRACT. The appearance of elliptic periodic orbits in families of n-dimensional smooth repelling billiard-like potentials that are arbitrarily steep is established for any finite n. Furthermore, the stability regions in the parameter space scale as a power-law in 1/n and in the steepness parameter. Thus, it is shown that even though these systems have a uniformly hyperbolic (albeit singular) limit, the ergodicity properties of this limit system are destroyed in the more realistic smooth setting. The consider… Show more
“…For the now so-called Sinai billiard system, he proved that it has positive measure-theoretic (Kolmogorov-Sinai) entropy and is hyperbolic almost everywhere. See also [11,18,32,34] and also [14,26,[29][30][31] for relevant and recent results and references therein. In contrast, rays converge after reflecting from convex boundaries.…”
Section: Introduction and Main Resultsmentioning
confidence: 97%
“…The hard scatterer case can be modelled by considering the limit lim →0 V ρ (·, ), cf. [30,31]. This limiting case describes the Sinai billiard on the two torus with a circular scatterer of diameter ρ, and in this case the lower bound goes to infinity at the rate −2 ln ρ + O(1) as ρ goes to zero.…”
Section: Remark 12mentioning
confidence: 95%
“…a soft scatterer, cf. [14,30,31]. The author showed that the lower bound can be made arbitrarily large provided that ρ and are sufficiently small.…”
We present in this paper an approach to studying the topological entropy of a class of billiard systems. In this class, any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers. Combining the concept of anti-integrable limit with the theory of Lyusternik-Shnirel'man, we show that a billiard system in this class generically admits a set of non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The anti-integrable limit is the singular limit when scatterers shrink to points. In order to get around the singular limit and so as to apply the implicit function theorem, on auxiliary circles encircling these scatterers we define a length functional whose critical points are well-defined at the anti-integrable limit and give rise to billiard orbits when the scatterers are not points. Consequently, we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small.
“…For the now so-called Sinai billiard system, he proved that it has positive measure-theoretic (Kolmogorov-Sinai) entropy and is hyperbolic almost everywhere. See also [11,18,32,34] and also [14,26,[29][30][31] for relevant and recent results and references therein. In contrast, rays converge after reflecting from convex boundaries.…”
Section: Introduction and Main Resultsmentioning
confidence: 97%
“…The hard scatterer case can be modelled by considering the limit lim →0 V ρ (·, ), cf. [30,31]. This limiting case describes the Sinai billiard on the two torus with a circular scatterer of diameter ρ, and in this case the lower bound goes to infinity at the rate −2 ln ρ + O(1) as ρ goes to zero.…”
Section: Remark 12mentioning
confidence: 95%
“…a soft scatterer, cf. [14,30,31]. The author showed that the lower bound can be made arbitrarily large provided that ρ and are sufficiently small.…”
We present in this paper an approach to studying the topological entropy of a class of billiard systems. In this class, any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers. Combining the concept of anti-integrable limit with the theory of Lyusternik-Shnirel'man, we show that a billiard system in this class generically admits a set of non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The anti-integrable limit is the singular limit when scatterers shrink to points. In order to get around the singular limit and so as to apply the implicit function theorem, on auxiliary circles encircling these scatterers we define a length functional whose critical points are well-defined at the anti-integrable limit and give rise to billiard orbits when the scatterers are not points. Consequently, we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small.
“…By this approach, to better understand systems with very steep potentials at the domain's boundary, one studies the limit system in which the steep part is replaced by impacts. Once the dynamics under the HIS are known, one establishes which of its features persist [27,17] and how those which do not persist bifurcate [30,28].…”
A class of Hamiltonian impact systems exhibiting smooth near integrable behavior is presented. The underlying unperturbed model investigated is an integrable, separable, 2 degrees of freedom mechanical impact system with effectively bounded energy level sets and a single straight wall which preserves the separable structure. Singularities in the system appear either
“…A partial answer is provided by the ergodic hy- * Electronic address: kushals@iiserb.ac.in pothesis which states that all accessible microstates of a given system are equiprobable over sufficiently long periods of time [4,5]. However, there are very few dynamical systems which have been actually proven to be ergodic [6,7,[9][10][11][12]. And even for ergodic systems, the time required for ergodization may be so long that it may be practically irrelevant.…”
Statistical equilibration of energies in a slow-fast system is a fundamental open problem in physics. In a recent paper, it was shown that the equilibration rate in a springy billiard can remain strictly positive in the limit of vanishing mass ratio (of the particle and billiard wall) when the frozen billiard has more than one ergodic components [Proc. Natl. Acad. Sci. USA 114, E10514 (2017)]. In this paper, using the model of a springy Sinai billiard, it is shown that this can happen even in the case where the frozen billiard has a single ergodic component, but when the time of ergodization in the frozen system is much longer than the time of equilibration. It is also shown that as the size of the disc in the Sinai billiard is increased from zero, thereby leading to a decrease in the time required for ergodization in the frozen system, the system undergoes a smooth phase transition in the equilibration rate dependence on mass ratio.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.