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200-0229-1038-9200, BLZ Dakteur Mitwirkte, Als Herausgeber Die 12000, BIC/SWIFT-Code BKAUATWW) Internationale Mathematische Nachrichten International Mathematical News Nouvelles Mathe´matiques Internationales Die IMN wurden 1947 von R. Inzin- Bezug: ger als Nachrichten der Mathematischen ” Gesellschaft in Wien“ gegru¨ndet. 1952 Die IMN erscheinen dreimal ja¨hrlich und ¨ wurde die Zeitschrift in Internationale werden von den Mitgliedern der Oster- ” Mathematische Nachrichten“ umbenannt reichischen Mathematischen Gesellschaft und war bis 1971 offizielles Publikati- bezogen. onsorgan der Internationalen Mathema- Jahresbeitrag: 20,– ” tischen Union“. Bankverbindung: Konto Nr. 229-103- Von 1953 bis 1977 betreute W. Wunder- 892-00 der Bank Austria–Creditanstalt lich, der bereits seit der Gru¨ndung als Re- (IBAN AT83-1200-0229-1038-9200, BLZ dakteur mitwirkte, als Herausgeber die 12000, BIC/SWIFT-Code BKAUATWW). IMN. Die weiteren Herausgeber waren H. Vogler (1978–79), U. Dieter (1980– 81, 1984–85), L. Reich (1982–83) und P. Flor (1986–99). Herausgeber: O¨ sterreichische Mathematische Gesell- schaft, Wiedner Hauptstraße 8–10/104, A-1040 Wien. e-mail [email protected], http://www.oemg.ac.at/ Redaktion: M. Drmota (TU Wien, Herausgeber) Eigentu¨mer, Herausgeber und Verleger: U. Dieter (TU Graz) O¨ sterr. Math. Gesellschaft. Satz: O¨ sterr. J. Wallner (TU Wien) Math. Gesellschaft. Druck: Grafisches R. Winkler (TU Wien) Zentrum, Wiedner Hauptstraße 8–10, 1040 Wien. Sta¨ndige Mitarbeiter der Redaktion: ¨ C. Binder (TU Wien) c 2005 Osterreichische Mathematische R. Mlitz (TU Wien) Gesellschaft, Wien. K. Sigmund (Univ. Wien) ISSN 0020-7926 O¨ sterreichische Mathematische Gesellschaft Gegru¨ndet 1903 Beirat: A. Binder (Linz) Sekretariat: C. Christian (Univ. Wien) TU Wien, Institut 104, U. Dieter (TU Graz) Wiedner Hauptstr. 8–10, A 1040 Wien. G. Gottlob (TU Wien) Tel. +43-1-58801-11823 P. M. Gruber (TU Wien) email: [email protected] G. Helmberg (Univ. Innsbruck) H. Heugl (Wien) E. Hlawka (TU Wien) Vorstand: W. Imrich (MU Leoben) M. Koth (Univ. Wien) R. Tichy (TU Graz): Vorsitzender W. Kuich (TU Wien) W. Schachermayer (TU Wien): R. Mlitz (TU Wien) Stellvertretender Vorsitzender W. Mu¨ller (Klagenfurt) M. Drmota (TU Wien): W. G. Nowak (Univ. Bodenkult. Wien) Herausgeber der IMN N. Rozsenich (Wien) M. Oberguggenberger (Univ. Inns- F. Schweiger (Univ. Salzburg) bruck): Schriftfu¨hrer K. Sigmund (Univ. Wien) I. Fischer (Univ. Klagenfurt): H. Sorger (Wien) Stellvertretende Schriftfu¨hrerin H. Stachel (TU Wien) H. Pottmann (TU Wien): H. Strasser (WU Wien) Kassier G. Teschl (Univ. Wien) F. Rendl (Univ. Klagenfurt): H. Troger (TU Wien) Stellvertretender Kassier W. Wurm (Wien) G. Teschl (Univ. Wien): Web-Beauftragter (kooptiert) Vorstand, Sektions- und Kommissi- onsvorsitzende geho¨ren statutengema¨ß Vorsitzende der Sektionen dem Beirat an. und Kommissionen: Mitgliedsbeitrag: L. Reich (Graz) A. Ostermann (Innsbruck) Jahresbeitrag: 20,– H. Kautschitsch (Klagenfurt) Bankverbindung: Konto Nr. 229-103- G. Larcher (Linz) 892-00 der Bank Austria–Creditanstalt P. Hellekalek (Salzburg) (IBAN AT83-1200-0229-1038-9200, C. Schmeiser (Wien) BLZ 12000, BIC BKAUATWW). R. Geretschlager ¨ (Lehrersektion) http://www.oemg.ac.at/ W. Schloglmann ¨ (Didaktik- email: [email protected] kommission) Internationale Mathematische Nachrichten International Mathematical News Nouvelles Mathe´matiques Internationales Nr. 200 (59. Jahrgang) Dezember 2005 Inhalt Klaus Schmidt: Algebra, arithmetic and multi-parameter ergodic theory . 1 Wolfgang Lu¨ck und Vasco Alexander Schmidt: Interview mit Friedrich Hir- zebruch . 23 Wolfgang Schlo¨glmann: Gedanken zum Lehramtsstudium im Unterrichts- fach Mathematik . 35 Johann Cigler: Erinnerungen . 43 Karl Perktold, Ju¨rgen Pu¨ngel, Robert F. Tichy: Helmut Florian 1924–2005 49 Buchbesprechungen . 53 Internationale Mathematische Nachrichten . 63 Nachrichten der O¨ sterreichischen Mathematischen Gesellschaft . 67 Kunden der sta¨dtischen Autobusse in San Francisco konnten sich im Februar 2005 u¨ber ein Ra¨tsel a¨hnlich dem auf der Titelseite abgebildeten den Kopf zerbrechen. Fu¨r die Lo¨sung eines der monatlich vom Mathematical Sciences Research In- stitute in Berkeley gestalteten Puzzles on Wheels waren US$ 100,– zu gewinnen (siehe http://www.msri.org/pow). Internat. Math. Nachrichten Nr. 200 (2005), 1–21 Algebra, arithmetic and multi-parameter ergodic theory Klaus Schmidt University of Vienna 1 Introduction While classical ergodic theory deals largely with single ergodic transformations or flows (i.e., with actions of N;Z;R+ or R on measure spaces), many of the lattice models in statistical mechanics (such as dimer models) have multi-dimensional symmetry groups: they carry actions of Zd or Rd with d > 1. However, the transi- tion from Z- or R-actions to multi-parameter ergodic theory presents considerable difficulties, even if one restricts attention to actions of Zd with d 1 (as we shall do throughout this article). ≥ To illustrate this point, compare the classical theory of topological Markov chains (cf. e.g. [31]) with the complexities and undecidability problems arising in the study of cellular automata and more general multi-dimensional shifts of finite type (cf. [3], [49] or [24]). Even if undecidability is not an issue, multi-dimensional shift of finite type exhibit a markedly more complicated behaviour than their clas- sical relatives (cf. e.g. [10, 11, 36, 42]). Another feature of the transition from d = 1 to d > 1 is that smooth Zd-actions with d > 1 on compact manifolds have zero entropy, since individual elements of Zd act with finite entropy. The powerful ideas and tools of smooth ergodic theory are thus of limited use for Zd-actions. Furthermore, smooth Zd-actions are not exactly abundant: all known examples arise from `algebraic' constructions (com- muting group translations, commuting automorphisms of finite-dimensional tori or solenoids, or actions of Cartan subgroups of semisimple Lie groups on homo- geneous spaces). Again one should compare this with the richness of examples in classical smooth ergodic theory which contributes so much to the appeal of the subject. ISSN 0020-7926 c 2005 O¨ sterr. Math. Gesellschaft Making a virtue out of necessity, let us briefly turn to commuting automorphisms of finite-dimensional tori. Toral automorphisms are among the longest and most intensively studied measure-preserving transformations (their investigation con- tributed much to the formulation and understanding of fundamental dynamical concepts like hyperbolicity and geometrical notions of entropy), and it came as a considerable surprise when Hillel Furstenberg [19] proved in 1967 that unex- pected things may happen if one studies not one, but two commuting toral maps: he showed that the only closed infinite subset of the circle T = R=Z which is si- multaneously invariant under multiplication by 2 and by 3 is the circle itself (this is a statement about commuting surjective homomorphisms of T, but it has an immediate extension to commuting automorphisms of the 6-adic solenoid). In contrast, each of the two maps consisting of multiplication by 2 and by 3, re- spectively, is very easily seen to have many infinite closed invariant subsets. In connection with this result Furstenberg asked the famous – and still unanswered – question whether Lebesgue measure is the only nonatomic probability measure on T which is simultaneously invariant under multiplication by 2 and by 3. A partial answer to Furstenberg's question was given by D. Rudolph in [39], where he showed that Lebesgue measure is the only nonatomic probability measure on T which is ergodic under the N2-action generated by multiplication by 2 and by 3, and which has positive entropy under at least one of these maps. The results by Furstenberg on invariant sets and by Rudolph on invariant measures have sub- sequently been extended to commuting toral and solenoidal automorphisms by D. Berend [1, 2], A. Katok and R. Spatzier [22] and M. Einsiedler and E. Linden- strauss in [15]. In 1978, Ledrappier [30] presented another surprising example: two commuting automorphisms of a compact abelian group such that the Z2-action generated by them is mixing, but not mixing of higher order (the problem whether there exists a single finite measure preserving transformation with this property is one of the famous unresolved questions in ergodic theory). These examples by Furstenberg and Ledrappier sparked off a systematic investiga- tion of Zd-actions by commuting automorphisms of compact groups (which will be referred to as algebraic Zd-action throughout this article). A key ingredient of this study, which began in 1989–1990 with the papers [25], [32], [40] and [41], is the connection of algebraic Zd-actions with commutative algebra and arithmeti- cal algebraic geometry. By combining ideas and methods from these areas with standard tools of ergodic theory one can obtain a great deal of insight into these actions and effectively resolve some rather difficult problems like higher order mixing, entropy calculations or the Bernoulli property. For reasons of space I will not discuss the very intriguing rigidity properties of algebraic Zd-actions (such as scarcity of invariant probability measures and isomorphism rigidity). The inter- ested reader can pursue these topics in the papers [4, 5, 6, 15, 21, 22, 27]. Instead I will focus on the many links between dynamics, algebra and arithmetic which 2 become apparent in the investigation of these actions. These notes are an expanded and updated version of the lecture [44] by the author at the Third European Congress of Mathematics in Barcelona. I would like to end this introduction by thanking Michael Baake for bringing the reference [50] to my attention. 2 Algebraic Zd-actions and their dual modules Let a: n an be an action of Zd; d 1, by continuous automorphisms of a 7! ≥ compact abelian group X with Borel field BX and normalized Haar measure lX . If b is a second algebraic Zd-action on a compact abelian group Y, then b is an algebraic factor of a if there exists a continuous surjective group homomorphism f: X Y with −! n n f a = b f (2.1) · · for every n Zd. The actions a and b are finitely equivalent if each of them is a finite-to-on2e algebraic factor of the other.
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