Introduction to Geometry. By H. S. M. Coxeter. Pp. xiv, 443. 1961. (John Wiley)

We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.

Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow

There are numerous physical situations in which a hole or leak is introduced in an otherwise closed chaotic system. The leak can have a natural origin, it can mimic measurement devices, and it can also be used to reveal dynamical properties of the closed system. A unified treatment of leaking systems is provided and applications to different physical problems, in both the classical and quantum pictures, are reviewed. The treatment is based on the transient chaos theory of open systems, which is essential because real leaks have finite size and therefore estimations based on the closed system differ essentially from observations. The field of applications reviewed is very broad, ranging from planetary astronomy and hydrodynamical flows to plasma physics and quantum fidelity. The theory is expanded and adapted to the case of partial leaks (partial absorption and/or transmission) with applications to room acoustics and optical microcavities in mind. Simulations in the lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on family of billiards illustrate the main text. Regarding billiard dynamics, it is emphasized that a correct discrete-time representation can be given only in terms of the so-called true-time maps, while traditional Poincar\'e maps lead to erroneous results. Perron-Frobenius-type operators are generalized so that they describe true-time maps with partial leaks.

Leaking chaotic systems

This is a survey paper on various topics concerning self-similar groups and branch groups with a focus on those notions and problems that are related to a 3-generated torsion 2 group of intermediate growth G, constructed by the author in 1980, and its generalizations Gω, ω ∈ {0, 1, 2}ℕ.

Solved and Unsolved Problems Around One Group

Asymptotic aspects of Schreier graphs and Hanoi Towers groups

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3.

On a free group of transformations defined by an automaton

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3

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The objective of the paper is to study properties of periodic geodesics on translation surfaces that hold for generic elements of the moduli space of translation surfaces. 2000 Mathematics Subject Classification: 37C27.

Periodic geodesics on generic translation surfaces

The paper is concerned with the space of the marked Schreier graphs of the Grigorchuk group and the action of the group on this space. In particular, we describe an invariant set of the Schreier graphs corresponding to the action on the boundary of the binary rooted tree and dynamics of the group action restricted to this invariant set.

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Notes on the Schreier graphs of the Grigorchuk group

We construct automata over a binary alphabet with 2n states, n ≥ 2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free products of cyclic groups of order two via such automata.

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Yaroslav Vorobets

Papers

On a free group of transformations defined by an automaton

On a free group of transformations defined by an automaton

Periodic geodesics on generic translation surfaces

Notes on the Schreier graphs of the Grigorchuk group

Automata over a binary alphabet generating free groups of even rank