# Singularities and nonhyperbolic manifolds do not coincide

@article{Simnyi2012SingularitiesAN, title={Singularities and nonhyperbolic manifolds do not coincide}, author={Nandor Sim{\'a}nyi}, journal={arXiv: Dynamical Systems}, year={2012} }

We consider the billiard flow of elastically colliding hard balls on the flat $\nu$-torus ($\nu\ge 2$), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli mixing property) of all such systems, i.e. the verification of the Boltzmann-Sinai Ergodic Hypothesis.

## 7 Citations

Symbolic dynamics for nonuniformly hyperbolic maps with singularities in high dimension

- Mathematics
- 2020

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…

Multidimensional hyperbolic billiards

- Mathematics
- 2017

The theory of planar hyperbolic billiards is already quite well developed by having also achieved spectacular successes. In addition there also exists an excellent monograph by Chernov and Markarian…

Further Developments of Sinai’s Ideas: The Boltzmann–Sinai Hypothesis

- MathematicsThe Abel Prize
- 2019

In this chapter we present a brief survey of the rich and manifold developments of Sinai’s ideas, dating back to 1963, concerning his exact mathematical formulation of Boltzmann’s original ergodic…

The Abel Prize 2013-2017

- The Abel Prize
- 2019

We review some of the most remarkable results obtained by Ya.G. Sinai and collaborators on the difficult problems arising in the theory of the Navier– Stokes equations and related models. The survey…

What mathematical billiards teach us about statistical physics

- Mathematics, Physics
- 2020

We survey applications of the theory of hyperbolic (and to a lesser extent non hyperbolic) billiards to some fundamental problems of statistical physics and their mathematically rigorous derivations…

Diffusion in the Lorentz gas

- Physics, Mathematics
- 2014

The Lorentz gas, a point particle making mirror-like reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for…

International Association of Mathematical Physics Μ Φ U Invitation

- 2012

Dear IAMP Members, according to Part I of the By-Laws we announce a meeting of the IAMP General Assembly. It will convene on Monday August 3 in the Meridian Hall of the Clarion Congress Hotel in…

## References

SHOWING 1-10 OF 25 REFERENCES

Nonuniformly hyperbolic K-systems are Bernoulli

- Mathematics
- 1996

We prove that those non-uniformly hyperbolic maps and flows (with singularities) that enjoy the K-property are also Bernoulli. In particular, many billiard systems, including those systems of hard…

Conditional proof of the Boltzmann-Sinai ergodic hypothesis

- Mathematics
- 2006

AbstractWe consider the system of N (≥ 2) elastically colliding hard balls of masses m1,…,mN and radius r on the flat unit torus
$\mathbb{T}^{\nu}$
, ν≥2. We prove the so called Boltzmann-Sinai…

Hard ball systems are completely hyperbolic

- Mathematics
- 1997

We consider the system of N (\ge 2) elastically colliding hard balls with masses m_1,..., m_N, radius r, moving uniformly in the flat torus T_L^{\nu}= R^\nu/L \cdot Z^\nu, \nu \ge 2. It is proved…

Hard ball systems are completely hyperbolic

- Mathematics
- 1999

We consider the system of N (‚ 2) elastically colliding hard balls with masses m1;:::, mN , radius r, moving uniformly in the ∞at torus T ” =R ” =L¢ Z ” , ” ‚ 2. It is proved here that the relevant…

The K-property of three billiard balls

- Mathematics
- 1991

Sinai's strengthened version of the ergodic hypothesis is proved for three billiard balls on the v-dimensional torus: On connected components of the submanifold of the phase space specified by the…

The K-property ofN billiard balls II. Computation of neutral linear spaces

- Mathematics
- 1992

SummaryThe present second part of the article “The K-property ofN billiard balls” finishes the proof of the Kolmogorov mixing property for hard ball systems in tori, provided that the number of balls…

Dynamical systems with elastic reflections

- Mathematics
- 1970

In this paper we consider dynamical systems resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is…

The complete hyperbolicity of cylindric billiards

- Mathematics, PhysicsErgodic Theory and Dynamical Systems
- 2002

The connected configuration space of a so-called cylindric billiard system is a flat torus minus finitely many spherical cylinders. The dynamical system describes the uniform motion of a point…

On the Bernoulli nature of systems with some hyperbolic structure

- Mathematics
- 1998

It is shown that systems with hyperbolic structure have the Bernoulli property. Some new results on smooth cross-sections of hyperbolic Bernoulli flows are also derived. The proofs involve an…

A “Transversal” Fundamental Theorem for semi-dispersing billiards

- 1990

For billiards with a hyperbolic behavior, Fundamental Theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds. A “Transversal”…