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Dynamical systems of a billiard type are a fundamental notion relevant for the understanding of numerous phenomena observed in statistical mechanics, Hamiltonian dynamics, nonlinear physics, and others. By means of billiard models, the principal ideas as Boltzmann ergodic hypothesis, related to the foundation of statistical physics, and deterministic diffusion gained more deep insight. However, more sophisticated and realistic results may be obtained if billiards are considered with time-dependent boundaries. Such kind of billiard systems represent a natural generalization of mathematical billiards and more adequately reflects the observed physical phenomena. This Focus Issue presents recent progress in this area with contributions on new ways and specific model studies.

In the last four decades of the twentieth century, mathematical billiards and related fields became one of the most active research areas in statistical mechanics and theory of dynamical systems. However, the history of billiard problems started in 1927 with a remarkable paper by Birkhoff [

A billiard dynamical system is generated by the free motion of a point mass particle (billiard ball) in some region with a piecewise-smooth boundary and the condition of the elastic collision from this boundary. If the boundary in the collision point is smooth, then the billiard ball reflects from it in such a way that the velocity tangent component remains constant, and the normal component changes its sign. If the ball hits a corner of the billiard table, then its dynamics is not determined.

Nowadays, the main focus in the study of billiard problems is moving from the investigations of billiards with the fixed boundary to time-dependent billiards. Indeed, the Lorentz gas has been proposed for the description of the motion of electrons between heavy ions in the lattice of metals. In the reality, however, ions should weakly oscillate near their equilibrium state. Moreover, some important problems of mathematical physics can be described by nonstationary billiard models (see [

For the most part, investigations of classical time-dependent billiards concerned two main questions: descriptions of their statistical properties and the study of trajectories for which the particle velocity grows indefinitely. This problem is related to the unbounded increase of energy in periodically forced Hamiltonian systems and known as Fermi acceleration [

It is obvious that in this context dynamical properties of billiards play a principal role: if it is chaotic, then the boundary perturbation may lead to the particle acceleration. In papers [

The next physical generalization of billiard ideas is to consider the corresponding dynamical systems with inelastic collisions. These models admits more realistic analysis of some natural phenomena. For example, they allow us to study the presence and lack of Fermi acceleration in certain billiard-like models.

The contributions to this focus issue can be grouped into three main parts:

Hamiltonian dynamics and related fields;

dynamics of time-dependent billiard-like models;

applications.

The first part on

The second part contains in-depth studies of specific models of

The third part on

We do hope that the presented papers will be interested in a wide audience of readers.

A. Loskutov gratefully acknowledges FAPESP and Departamento de Estatística, Matemática Aplicada e Computação (Brazil) for the summer grant during his stay in the Universidade Estadual Paulista, Rio Claro, Brazil. E. D. Leonel acknowledges the financial support from CNPq, FAPESP, CAPES, FUNDUNESP and Pró Reitoria de Pesquisa (PROPe, UNESP) and Pró Reitoria de Pós-Graduação (PROPG, UNESP). They are grateful to Professors Makoto Yoshida (UNESP, Brazil), Jafferson Silva (UFMG, Brazil), and Leonid Bunimovich (Georgia Institute of Tecnhology, USA), and all the contributors of the past International Meeting “Billiards'09” (Águas de Lindóia, Brazil), as well as the authors for their papers sent to the present issue.