Circle Decompositions of Surfaces

We determine which connected surfaces can be partitioned into topological circles. There are exactly seven such surfaces up to homeomorphism: those of finite type, of Euler characteristic zero, and with compact boundary components. As a byproduct, we get that any circle decomposition of a surface is upper semicontinuous.Comment: 9 pages, final version, to appear in Topology and its Applications (2010). A few missprints have been correcte

Proof of the Boltzmann-Sinai Ergodic Hypothesis for Typical Hard Disk Systems

We consider the system of $N$ ($\ge2$) hard disks of masses $m_1,...,m_N$ and radius $r$ in the flat unit torus $\Bbb T^2$. We prove the ergodicity (actually, the B-mixing property) of such systems for almost every selection $(m_1,...,m_N;r)$ of the outer geometric parameters.Comment: 58 page

Singularities and nonhyperbolic manifolds do not coincide

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.Comment: Final version, to appear in Nonlinearit

The characteristic exponents of the falling ball model

We study the characteristic exponents of the Hamiltonian system of $n$ ($\ge 2$) point masses $m_1,\dots,m_n$ freely falling in the vertical half line $\{q|\, q\ge 0\}$ under constant gravitation and colliding with each other and the solid floor $q=0$ elastically. This model was introduced and first studied by M. Wojtkowski. Hereby we prove his conjecture: All relevant characteristic (Lyapunov) exponents of the above dynamical system are nonzero, provided that $m_1\ge\dots\ge m_n$ (i. e. the masses do not increase as we go up) and $m_1\ne m_2$

Rotation sets of billiards with one obstacle

We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets

A polinomális-leképezés módszer és annak alkalmazásai az analízisban = The polynomial mapping method and its applications analysis

A pályázat által támogatott kutatások a komplex függvénytanhoz, a sorelmélethez és a potenciálelmélethez kapcsolódnak. Pályázati támogatással egy könyv, 37 tudományos közlemény és 4 doktori disszertáció született. A főbb kutatási területek a következők voltak: - harmonikus mértékek és Green függvények becslése végtelenszer összefüggő tartományokon - a polinom-inverz kép módszer és polinom-egyenlőtlenségek - a polinom-inverz kép módszer és polinom-approximáció - Green függvények Hölder folytonossága - ortogonális polinomok - beágyzási tételek - sorelmélet - statisztikus konvergencia - Fourier-sorok - approximációelmélet Ezen kérdésekkel kapcsolatos kutatásaink és publikációink jelentős mértékben gazdagítottak e területekre vonatkozó ismereteinket. | The research supported by the project is connected with function theory, the theory of series and potential theory. One book, 37 scientific publications and 4 PhD dissertations were published with the support of the project. The major areas were the following: - harmonic measures and Green's functions on infinitely connected domains - the polynomial-inverse image method and polynomial inequalities - the polynomial-inverse image method and polynomial approximation - Holder continuity of Green's functions - orgthogonal polynomials - imbedding theorems - statistical convergence - Fourier series - approximation theory The supported research and publications in connection with these questions considerably enhanced our understanding of these areas