Non-Uniformly Hyperbolic Dynamical Systems with Singularities

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This project is devoted to a special class of chaotic dynamical systems,namely hyperbolic mathematical billiards. They serve as the prototype examplesof non-uniformly hyperbolic dynamical systems with singularities. Such systemsplay an increasingly important role in the rigorous mathematical foundation ofstatistical physics, so that the study of their chaotic (i. e. mixing)properties is getting more and more physical relevance. The project mainlyfocuses on a fundamental conjecture regarding this family of dynamicalsystems, namely the celebrated "Boltzmann-Sinai Ergodic Conjecture", whichstates that any finite system of (totally elastic) hard spheres moving on aflat torus is fully hyperbolic and ergodic, of course, on the level set of itstrivial first integrals. The proof of this conjecture (in its full generality)has been so far notoriously withstanding any attack against it. The firstmajor part of the present proposal directly targets this conjecture. Thesecond and fourth parts are blueprints for further research in this directionby generalizing the original Boltzmann-Sinai Conjecture to cylindric billiards(mathematical billiards with cylindric scatterers) and billiards in physicallymore relevant containers, like rectangular boxes. The third part of the projectaims at the biggest open question in the topic of Wojtkowski's one-dimensionalfalling balls: Wojtkowski's still unsolved conjecture on the full hyperbolicityof the falling ball system with nonincreasing masses. (And such that not allmasses are the same, of course.) Beside these, the question of ergodicity(possibly, under the condition that a strictly concave potential acts) is alsoposed and targeted. The foundation of statistical physics (like heat theory, dynamical theoryof fluids and gases) took place in the last third of the 18th century, mainlyby the groundbreaking works of Boltzmann and Helmholz. That foundation was,however, based upon a strong hypothesis made by Boltzmann himself. Thathypothesis claims that any physical system with a huge number of interactingparticles (like molecules) has the property that for any fixed total energyand initial state, the system will evolve to any other state with the sameenergy. Although this conjecture, if taken literally, mathematically cannothappen, yet the precise mathematical formalism and its rigorous verificationfor different models of statistical physics bears a particular importance tothe understanding the physics of the surrounding world.