arXiv subjects

Nandor Simanyi

Publications and source records attributed to Nandor Simanyi.

At least 19 recordsLinked to original sources

arXiv2025

Asymptotic Homotopical Complexity of an Infinite Sequence of Dispersing $2D$ Billiards

We investigate the large scale chaotic, topological structure of the trajectories of an infinite sequence of dispersing, hence ergodic, $2D$ billiards with the configuration space $Q_n=\mathbb{T}^2 \setminus \bigcup_{i=0}^{n-1} D_i$, where the scatterers $D_i$ ($i=0,1,\dots,n-1$) are disks of radius $r<<1$ centered at the points $(i/n, 0)$ mod $\mathbb{Z}^2$. We get effective lower and upper radial bounds for the rotation set $R$. Furthermore, we also prove the compactness of the admissible rotation set $AR$ and the fact that the rotation vectors $v$ corresponding to admissible periodic orbits form a dense subset of $AR$. We also obtain asymptotic lower and upper estimates for the sequence $h_{top}(n)$ of topological entropies and precise asymptotic formulas for the metric entropies $h_{\mu}(n,r)$.

math.DS↗

arXiv2024

Proof of Wojtkowski's Falling Particle Conjecture

In this paper we present an unconditional proof of Wojtkowski's Ergodicity Conjecture for almost every system of 1D perfectly elastic balls falling down in a half line under constant gravitational acceleration. Namely, by introducing a new algebraic approach, we prove that almost every such system is (completely hyperbolic and) ergodic.

math.DS↗

arXiv2022

Conditional Proof of the Ergodic Conjecture for Falling Ball Systems

In this paper we present a conditional proof of Wojtkowski's Ergodicity Conjecture for the system of 1D perfectly elastic balls falling down in a half line under constant gravitational acceleration. Namely, we prove that almost every such system is (completely hyperbolic and) ergodic, by assuming the transversality between different singularities and between singularities and stable (unstable) invariant manifolds.

math.DS↗

arXiv2015

Further Developments of Sinai's Ideas: The Boltzmann-Sinai Hypothesis

In 1963 Ya. G. Sinai formulated a modern version of Boltzmann's ergodic hypothesis, what we now call the ``Boltzmann-Sinai Ergodic Hypothesis'': The billiard system of $N$ ($N\ge 2$) hard balls of unit mass moving on the flat torus $\mathbb{T}^ν=\mathbb{R}^ν/\mathbb{Z}^ν$ ($ν\ge 2$) is ergodic after we make the standard reductions by fixing the values of trivial invariant quantities. It took fifty years and the efforts of several people, including Sinai himself, until this conjecture was finally proved. In this short survey we provide a quick review of the closing part of this process, by showing how Sinai's original ideas developed further between 2000 and 2013, eventually leading the proof of the conjecture.

math.DS↗

arXiv2013

Singularities and nonhyperbolic manifolds do not coincide

We consider the billiard flow of elastically colliding hard balls on the flat $ν$-torus ($ν\ge 2$), 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.

math.DS↗

arXiv2012

Stable regimes for hard disks in a channel with twisting walls

We study a gas of $N$ hard disks in a box with semi-periodic boundary conditions. The unperturbed gas is hyperbolic and ergodic (these facts are proved for N=2 and expected to be true for all $N\geq 2$). We study various perturbations by twisting the outgoing velocity at collisions with the walls. We show that the dynamics tends to collapse to various stable regimes, however we define the perturbations and however small they are.

math.DS↗

arXiv2011

Sums of squares and orthogonal integral vectors

Two vectors in $\BZ^3$ are called \emph{twins} if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers $M$ with the property that each integral vector with length $\sqrt{M}$ has a twin are called twin-complete. They are completely characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions. Throughout the paper we made a concerted effort to keep the exposition as elementary as possible.

math.NT↗

arXiv2010

Homotopical Complexity of 2D Billiard Orbits

Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. The billard trajectories, being curves, oftentimes getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e., the billiard flow on the 2-torus with a single, convex obstacle removed. This fundamental group is known to be the group $\textbf{F}_2$ freely generated by two elements, which is a heavily noncommutative, hyperbolic group in Gromov's sense. We define the homotopical rotation number and the homotopical rotation set for this model, and provide lower and upper estimates for the latter one, along with checking the validity of classically expected properties, like the density (in the homotopical rotation set) of the homotopical rotation numbers of periodic orbits. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(\textbf{F}_2)$ of all "ends" of the hyperbolic group $\textbf{F}_2$. An element of $\text{Ends}(\textbf{F}_2)$ describes the direction in (the Cayley graph of) the group $\textbf{F}_2$ in which the considered trajectory escapes to infinity, whereas the height function $t$ ($t \ge 0$) of the cone gives us the average speed at which this escape takes place. The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding $\sqrt{2}$, and any direction $e\in\text{Ends}(F_2)$ for the escape is feasible with any prescribed speed $s$, $0\leq s\leq \sqrt{2}/2$. This means that the radial upper and lower bounds for the rotation set $R$ are actually pretty close to each other.

math.DS↗

arXiv2010

The Boltzmann-Sinai Ergodic Hypothesis in Two Dimensions (Without Exceptional Models)

We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ in the flat unit torus $\Bbb T^ν$, $ν\ge2$. In the case $ν=2$ we prove (the full hyperbolicity and) the ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters, without exceptional values. In higher dimensions, for hard ball systems in $\Bbb T^ν$ ($ν\ge3$), we prove that every such system (is fully hyperbolic and) has open ergodic components.

math.DS↗

arXiv2010

The Boltzmann--Sinai Ergodic Hypothesis In Full Generality

In the ergodic theory of semi-dispersing billiards the Local Ergodic Theorem, proved by Chernov and Sinai in 1987, plays a central role. So far, all existing proofs of this theorem had to use an annoying global hypothesis, namely the almost sure hyperbolicity of singular orbits. (This is the so called Chernov--Sinai Ansatz.) Here we introduce some new geometric ideas to overcome this difficulty and liberate the proof from the tyranny of the Ansatz. The presented proof is a substantial generalization of my previous joint result with N. Chernov (which is a $2D$ result) to arbitrary dimensions. An important corollary of the presented ansatz-free proof of the Local Ergodic Theorem is finally completing the proof of the Boltzmann--Sinai Ergodic Hypothesis for hard ball systems in full generality.

math.DS↗

arXiv2010

Upgrading the Theorem on Local Ergodicity

We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the condition of the so called ``Ansatz'' can be dropped. That condition assumed that almost every singular phase point had a hyperbolic trajectory after the singularity. Having this condition dropped, the cited theorem becomes much stronger and easier to apply. At the end of the paper two immediate corollaries of this improvement are discussed: One of them is the (fully hyperbolic) Bernoulli mixing property of every hard disk system (D=2), the other one claims that the ergodic components of every hard ball system ($D\ge3$) are open.

math.DS↗

arXiv2010

Improvement of the Theorem on Local Ergodicity

We prove here that in the Theorem on Local Ergodicity for Semi-Dispersive Billiards (proved by N. I. Chernov and Ya. G. Sinai in 1987) the recently added condition (by P. Bálint, N. Chernov, D. Szász, and I. P. Tóth, in order to save this fundamental result) on the algebraic character of the smooth boundary components of the configuration space is unnecessary. Having saved the theorem in its original form by using additional ideas in the spirit of the initial proof, the result becomes stronger and it applies to a larger family of models.

math.DS↗

arXiv2010

The Boltzmann-Sinai Ergodic Hypothesis in Full Generality (Without Exceptional Models)

We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^ν$, $ν\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters, without exceptional values. The present proof does not use at all the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools of geometric analysis.

math.DS↗

arXiv2010

Homotopical Rotation Numbers of 2D Billiards

Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. Naturally, these creatures are living in the (commutative) vector space $\real^n$, if the toroidal billiard is given on the flat $n$-torus. The billard trajectories, being curves, oftentimes getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e., the billiard flow on the 2-torus with a single, convex obstacle removed. This fundamental group is known to be the group $\textbf{F}_2$ freely generated by two elements, which is a heavily noncommutative, hyperbolic group in Gromov's sense. We define the homotopical rotation number and the homotopical rotation set for this model, and provide lower and upper estimates for the latter one, along with checking the validity of classicaly expected properties, like the density (in the homotopical rotation set) of the homotopical rotation numbers of periodic orbits. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(\textbf{F}_2)$ of all ``ends'' of the hyperbolic group $\textbf{F}_2$. An element of $\text{Ends}(\textbf{F}_2)$ describes the direction in (the Cayley graph of) the group $\textbf{F}_2$ in which the considered trajectory escapes to infinity, whereas the height function $t$ ($t \ge 0$) of the cone gives us the average speed at which this escape takes place.

math.DS↗

arXiv2010

Unconditional Proof of the Boltzmann-Sinai Ergodic Hypothesis

We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^ν$, $ν\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.

math.DS↗

arXiv2007

An Effective Contraction Estimate in the Stable Subspaces of Phase Points in Hard Ball Systems

In this paper we prove the following result, useful and often needed in the study of the ergodic properties of hard ball systems: In any such system, for any phase point x with a non-singular forward trajectory and infinitely many connected collision graphs on that forward orbit, it is true that for any small number epsilon there is a stable tangent vector w of x and a large enough time t>>1 so that the vector w undergoes a contraction by a factor of less than epsilon in time t. Of course, the Multiplicative Ergodic Theorem of Oseledets provides a much stronger conclusion, but at the expense of an unspecified zero-measured exceptional set of phase points, and this is not sufficient in the sophisticated studies the ergodic properties of such flows. Here the exceptional set of phase points is a dynamically characterized set, so that it suffices for the proofs showing how global ergodicity follows from the localone.

math.DS↗

arXiv2005

A Note on the Size of the Largest Ball Inside a Convex Polytope

Let $m>1$ be an integer, $B_m$ the set of all unit vectors of $\Bbb R^m$ pointing in the direction of a nonzero integer vector of the cube $[-1, 1]^m$. Denote by $s_m$ the radius of the largest ball contained in the convex hull of $B_m$. We determine the exact value of $s_m$ and obtain the asymptotic equality $s_m\sim\frac{2}{\sqrt{\log m}}$.

math.MG↗

arXiv2003

Proof of the Ergodic Hypothesis for Typical Hard Ball Systems

We consider the system of $N$ ($\ge2$) hard balls with masses $m_1,...,m_N$ and radius $r$ in the flat torus $\Bbb T_L^ν=\Bbb R^ν/L\cdot\Bbb Z^ν$ of size $L$, $ν\ge3$. We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection $(m_1,...,m_N; L)$ of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case $ν=2$. The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic.

math.DS↗