Mathematics

We study a mechanical system that was considered by Boltzmann in 1868 in the context of the derivation of the canonical and microcanonical ensembles. This system was introduced as an example of ergodic dynamics, which was central to Boltzmann's derivation. It consists of a single particle in two dimensions, which is subjected to a gravitational attraction to a fixed center. In addition, an infinite plane is fixed at some finite distance from the center, which acts as a hard wall on which the particle collides elastically. Finally, an extra centrifugal force is added. We will show that, in the absence of this extra centrifugal force, there are two independent integrals of motion. Therefore the extra centrifugal force is necessary for Boltzmann's claim of ergodicity to hold.

Let g(z)= ??z 0 p(t)exp(q(t))dt+c where p,q are polynomials and c?�C , and let f be the function from Newton's method for g . We show that under suitable assumptions the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that f n (z) converges to zeros of g almost everywhere in C if this is the case for each zero of g ?��?. In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

The paper contains a rigorous proof of existence of symbolic dynamics chaos in the generalized Hénon map's 4th iterate H 4 , which was conjectured in the paper \textit{A 3D Smale Horseshoe in a Hyperchaotic Discrete-Time System} of Li and Yang, 2007. We prove also the uniform hyperbolicity of the invariant set with symbolic dynamics. The proofs are computer-assisted with the use of C++ library \textit{CAPD} for interval arithmetic, differentiation and integration.

Wojtkowski's system of N , N≥2 , falling balls is a nonuniformly hyperbolic smooth dynamical system with singularities. It is still an open question whether this system is ergodic. We contribute towards an affirmative answer, by proving the non-contraction property, conditioned by the assumption of strict unboundedness. For a certain mass ratio the configuration space can be unfolded to a billiard table where the daunting proper alignment condition is satisfied. We prove, that the aforementioned unfolded system with three degrees of freedom is ergodic.

We use a result of J. Mather on the existence of connecting orbits for compositions of monotone twist maps of the cylinder to prove the existence of connecting geodesics on the unit tangent bundle S T 2 of the 2-torus in regions without invariant tori.

The first aims of this work are to endorse the advent of finitely additive set functions as equilibrium states and the possibility to replace the metric entropy by an upper semi-continuous map associated to a general variational principle. More precisely, using methods from Convex Analysis, we construct for each generalized convex pressure function an upper semi-continuous entropy-like map (which, in the context of continuous transformations acting on a compact metric space and the topological pressure, turns out to be the upper semi-continuous envelope of the Kolmogorov-Sinai metric entropy), then establish a new abstract variational principle and prove that equilibrium states, possibly finitely additive, always exist. This conceptual approach provides a new insight on dynamical systems without a measure with maximal entropy, prompts the study of finitely additive ground states for non-uniformly hyperbolic maps and grants the existence of finitely additive Lyapunov equilibrium states for singular value potentials generated by linear cocycles over continuous maps. We further investigate several applications, including a new thermodynamic formalism for systems driven by finitely generated semigroup or countable sofic group actions. On the final pages of the manuscript we provide a list of open problems in a wide range of topics suggested by our main results.

We construct an example of a smooth convex function on the plane with a strict minimum at zero, which is real analytic except at zero, for which Thom's gradient conjecture fails both at zero and infinity. More precisely, the gradient orbits of the function spiral around zero and at infinity. Besides, the function satisfies the Lojasiewicz gradient inequality at zero.

The main result of this paper states that for a given countable system of data, there exists a countable iterated function system consisting of Rakotch contractions, such that its attractor is the graph of a fractal interpolation function corresponding to the given countable system of data.

We study a discrete-time dynamical system of wild mosquito population with parameters: β - the birth rate of adults; α - maximum emergence rete; μ>0 - the death rate of adults; γ - Allee effects. We prove that if γ??α(β?��? μ 2 then the mosquito population dies and if γ< α(β?��? μ 2 holds then extinction or survival of the mosquito population depends on their initial state.

We give a strong fundamental domain for the quotient of PG L 2 ( F q (( t −1 ))) by PG L 2 ( F q [t]) as a subset of distinct ordered triple points of P 1 ( F q (( t −1 ))) .