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Convex Analysisの二,三の進展について

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Introduction to the Modern Theory of Dynamical Systems

Chaos, fractional kinetics, and anomalous transport

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

https://chaos.if.uj.edu.pl/~karol/erratum10.pdf

Geometry of Quantum States

Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

1. Introduction. Translation surfaces naturally arise in the study of billiards in rational polygons (see [ZKa]). To any such polygon P , there corresponds a unique translation surface, S = S(P), such that the billiard flow in P is equivalent to the geodesic flow on S (see, e.g., [Gu2], [Gu3]). There is also a classical relation between translation surfaces and quadratic differentials on a Riemann surface S. Namely, each quadratic differential induces a translation structure on a finite puncturing of S or on a canonical double covering of S. Quadratic differentials have a natural interpretation as cotangent vectors to Teich-müller space, and this connection has proven useful in the study of billiards (see, e.g., [Ma2], [V1]). With a translation surface, S, one associates various algebraic and geometric objects: the induced affine structure of S and the group of affine diffeomorphisms, Aff(S); the holonomy homomorphism, hol : π 1 (S) → R 2 and the holonomy group Hol(S) = hol(π 1 (S)); the flat structure on S and the natural cell decompositions of its metric completion S. In the present paper, we study the relations between these objects, as well as relations among different translation surfaces. Our main focus is the group Aff(S) and the associated group of differentials, (S) ⊂ SL(2, R). The study of these groups began as part of W. Thurston's classification of surface diffeomorphisms in [Th2]. This study continued with the work of W. Veech in [V1] and [V2]. Veech produced explicit examples of translation surfaces S for which (S) is a nonarithmetic lattice. He showed that if (S) is a lattice, then the geodesic flow on S exhibits remarkable dynamical properties. For these reasons, we call (S) the Veech group of S, and if this group is a lattice, then we call S a Veech surface. We now describe the structure of the paper and our main results. In §2, we establish the setting. In particular, we recall the notion of a G-manifold and associated objects: the developing map, the holonomy homomorphism, and the holonomy group. We introduce the notion of the differential of a G-map with respect to a normal subgroup H ⊂ G. We also introduce the spinal triangulation, one of several cell decompositions canonically associated to a flat surface with cone points. In §3, we study G-coverings of G-manifolds. Given such a covering, p : X → Y , we characterize the group …

Affine mappings of translation surfaces: geometry and arithmetic

In the ten or so years since the publication of ref. 19 polygonal billiards have remained an active subject of research in the mathematics and physics literature. As a result, our understanding of the subject, although still far from being complete, is much better than it was ten years ago. This survey attempts to give a broad overview of the dynamics of billiards in polygons, with an emphasis on the material that was not in the literature in 1985. This is not a survey of the publications on polygonal billiards, and we apologize to authors whose work has not been included. The selection of topics has been strongly influenced by the personal taste of the author, and by space limitations. Thus, we do not discuss quantum polygonal billiards (in particular, quantum chaos; see, e.g., refs. 41, 44, and 45). We hope that somebody will write a survey on this important subject in the near future. A fascinating aspect of the subject is the interplay between the geometric shape of the billiard table (i.e., a planar curve) and the qualitative features of the billiard dynamics. Hence it is instructive to compare polygonal billiard tables with other classes of billiard tables. In particular, the smooth, strictly convex tables and the dispersing (or Sinai) tables produce strikingly different types of billiard dynamics, We recommend refs. 46 and 15 for a general overview of the subject and a comparative study of billiard dynamics for various types of billiard tables.

Billiards in polygons: Survey of recent results

Billiards in polygons

The phase space of the geodesic flow on an almost integrable polyhedral surface is foliated into a one-parameter family of invariant surfaces. The flow on a typical invariant surface is minimal. We associate with an almost integrable polyhedral surface its holonomy group which is a subgroup of the group of motions of the Euclidean plane. We show that if the holonomy group is discrete then the flow on an invariant surface is ergodic if and only if it is minimal.

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Billiards on almost integrable polyhedral surfaces

https://www.mis.mpg.de/preprints/2003/preprint2003_1.pdf

Eugene Gutkin

Papers

Affine mappings of translation surfaces: geometry and arithmetic

Billiards in polygons: Survey of recent results

Billiards in polygons

Billiards on almost integrable polyhedral surfaces

Convexity of the joint numerical range: topological and differential geometric viewpoints