DOCSLIB.ORG

Adele Groups, P-Adic Groups, Solenoids

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Adele Groups, P-Adic Groups, Solenoids (April 29, 2014) Adele groups, p-adic groups, solenoids Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/mfms/notes 2013-14/12 1 adeles.pdf] 1. Hensel's lemma 2. Metric definition of p-adic integers Zp and p-adic rationals Qp 3. Elementary/clumsy definitions of adeles A and ideles J 4. Uniqueness of objects characterized by mapping properties 5. Existence of limits 6. Zp and Zb as limits 7. Qp and A as colimits 8. Abelian solenoids (R × Qp)=Z[1=p] and A=Q 9. Non-abelian solenoids and SL2(Q)nSL2(A) Although we will also give the more typical descriptions, we also show that the adeles A, p-adic integers Zp, p-adic rationals Qp, and related objects are already right under our noses, if only we can see them. These things are not artificial or mere stylistic choices. It is unwise to ignore them. Several important technical issues lead to such things. One is closer examination of Hecke operators, for example on holomorphic modular forms: X TN f = fj2kγ (f holomorphic weight 2k for Γ = SL2(Z)) γ2XN where XN is integer-entry 2-by-2 matrices with determinant 1, with the weight-2k action az + b a b fj γ(z) = f (cz + d)−2k (where γ = ) 2k cz + d c d An obstacle to a straightforward proof of the self-adjointness of these operators with respect to the Petersson inner product Z 2k dx dy hf; F i = f(z) F (z) y 2 ΓnH y is that, although the sum over XN is a modular form of level one, the individual summands are typically not level one, but higher level, usually level N. That is, the most straightforward assertion we can rely upon is th that for γ 2 XN the γ summand is a modular form only for a b 1 0 Γ = f = mod Ng N c d 0 1 That is, the straightforward assertion is that fj2kγ 2kβ = fj2kγ (for β 2 ΓN ) To see how the sufficient condition β 2 ΓN arises, use the associativity of the action: −1 −1 fj2k(γβ) = fj2k(γβγ · γ) = fj2kγβγ 2kγ −1 Thus, fj2kγ is assured invariant under β when γβγ 2 Γ1, for f of level 1. For example, a b a b=N 1 0 γ γ−1 = (for γ = ) c d cN d 0 N 1 Paul Garrett: Adele groups, p-adic groups, solenoids (April 29, 2014) which is in Γ1 = SL2(Z) only for b = 0 mod N. Similarly, a b a bN N 0 γ γ−1 = (for γ = ) c d c=N d 0 1 is in Γ1 only for c = 0 mod N. Generally, with γ 2 XN , similar computations show that β 2 ΓN is the simplest sufficient condition for (fj2kγ)j2kβ = fj2kγ. To treat the summands in TN f directly, we might enlarge the space of modular forms to include those for ΓN for all N. The full family of quotients ΓN nH, indexed by positive integers N ordered by divisibility, is partly depicted in ::: / Γ8nH / Γ4nH (where ΓN z ! ΓM z for MjN) v: HH vv HH vv HH vv HH vv H# ::: / Γ12nH / Γ6nH / Γ2nH : H D G vv HH © G vv HH © GG vv HH©© GG vv ©HH GG vv ©© # # :::v / Γ9nH © / Γ3nH / Γ1nH : ©© ; D vv © w vv © ww vv ©© ww vv © ww vv © w :::v / Γ10nH / Γ5nH ; vv vv vv vv vv :::v / Γ7nH The poset [1] of positive integers ordered by divisibility is very interesting, but adds a further complication, so we might consider subfamilies corresponding to powers of a single prime, with their simpler partial order structures: ::: / Γp3 nH / Γp2 nH / ΓpnH / Γ1nH In both cases, we want a single object on which all the functions on all spaces Γpn nH live, with quotient [2] maps to every Γpn nH, fitting together so that all triangles commute in the diagram + * ) ( ??? ::: / Γp3 nH / Γp2 nH / ΓpnH / Γ1nH The corresponding diagram for the full family of ΓN nH is messier, but similar. The admittedly natural candidate H itself, with obvious quotient maps qN : H ! ΓN nH fails in several ways. First, although 2 2 functions f 2 L (ΓN nH) become functions on H by composing with qN , the function f ◦ qN is not in L (H) 2 with respect to the SL2(R)-invariant measure dx dy=y . Second, the collection of all functions on H is far too large in the sense that most functions are not invariant under any ΓN . Third, certainly the true object ??? fits into a diagram ::: / Γ 3 nH / Γ 2 nH / ΓpnH / Γ1nH p p 5 dI O v: k k I v k I v k k I v k I v k k ??? k [1] Poset is the common abbreviation for partially ordered set, meaning a set S with an ordering ≤ with expected properties: transitivity x ≤ y and y ≤ z implies x ≤ z, reflexivity x ≤ x, and x ≤ y and y ≤ x implies x = y, but possibly lacking the dichotomy requirement that either x ≤ y or y ≤ x. [2] For a diagram to commute means that the same outcomes are obtained no matter which route is followed throught the diagram. 2 Paul Garrett: Adele groups, p-adic groups, solenoids (April 29, 2014) with commuting triangles but it turns out that there is no map ??? ! H fitting into + * ) ( H ::: / Γ 3 nH / Γ 2 nH / ΓpnH / Γ1nH (fails ...) k p p 5 dI O v: k k I v k I v k k non−existent I v k I v k k ??? k We will see that the correct object ??? in the diagram is / 1 , * ( ' SL2(Z[ p ]) H × SL2(Qp) ::: / Γp3 nH / Γp2 nH / ΓpnH / Γ1nH where Qp is the field of p-adic numbers, the fraction field of the p-adic integers Zp, the latter often defined as the completion of Z with respect to the p-adic metric d(x; y) = jx − yjp with the p-adic norm ` −` jp · mjp = p (for integers m prime to p) 1 and Z[ p ] is the ring of rational numbers with denominators only allowed to be powers of p. The action of 1 γ 2 SL2(Z[ p ]) on H × SL2(Q2) is g(z × h) = gz × gh with linear fractional action on z 2 H and left matrix 1 multiplication on h 2 SL2(Qp). We will see that the diagonally-imbedded copy of SL2(Z[ p ]) is a discrete subgroup of SL2(R) × SL2(Qp) much as SL2(Z) is a discrete subgroup of SL2(R). We will see that this object does have the desired universal property, that for every other object X with a compatible family of maps (making triangles commute) ::: / Γp3 nH / Γp2 nH / ΓpnH / Γ1nH dI O : j5 I v j j I v j j I v j j I vj j X jv 1 there is a unique map X ! SL2(Z[ p ])n H × SL2(Q2) giving a commutative diagram 1 * ( ' SL2( [ ])n H × SL2( p) ::: /, Γ 3 nH / Γ 2 nH / ΓpnH / Γ1nH Z p Q p p 5 l dI O v; k k I v k k I v k unique I v k k I k k X kv This situation also reveals a natural action of the group SL2(Qp) on the collection of quotients Γpn nH. This is very far from obvious! Keeping in mind that H ≈ SL2(R)=SO(2; R), the full diagram using all ΓN 's has a similar universal object, 3 Paul Garrett: Adele groups, p-adic groups, solenoids (April 29, 2014) [3] with A the ring of rational adeles ::: /, Γ2nH w; GG ww GG ww GG ww GG + w + ) # SL2(Q)nSL2(A)=SO(2; R) ::: / Γ6nH / Γ3nH / Γ1nH YYYY ; YYYYYY ww YYYYYY ww YYYYYY ww YYYYYY w YYYYYY w ::: /, Γ5nH For many reasons we might want to get the group SO(2; R) out of the way, so that SL2(R) can act on the right as do the groups SL2(Qp), and we have a diagram - ::: / Γ2nSL2(R) 7 O ooo OOO ooo OOO ooo OOO , oo + * O' SL2(Q)nSL2(A) ::: / Γ6nSL2(R) / Γ3nSL2(R) / Γ1nSL2(R) ZZZZ 7 ZZZZZZZ ooo ZZZZZZZ oo ZZZZZZ ooo ZZZZZZZ oo ZZZZZ, o ::: / Γ5nSL2(R) This reveals the natural action of SL2(A) on the right on the family of quotients ΓN nSL2(R), incorporating an action of every group SL2(Qp) as well as SL2(R). We will prove the universal mapping property that, for every space X and compatible family of maps X ! ΓN nSL2(R), there is a unique map X ! SL2(Q)nSL2(A) fitting into a commutative diagram - ::: / Γ2nSL2(R) h3 o7 OO k i oo OO m ooo OOO o ooo OOO , r oo + * O' SL2(Q)nSL2(A) ::: /tΓ6nSL2(R) / Γ3nSL2(R) / Γ1nSL2(R) ] ZZZZ w ; 6 7 : ZZZZZZZ x m m oou ZZZZZZZz x m ooo ZZZZxZZZ m oo s } x ZZZZZmZZm oooq x m ZZZZZZ, o m::: / Γ5nSL2(Ro) unique £ x m m 2 n x m f f f l ¦ x m f f f j x m m f f i ¨ x m f f f g x mf f f d f X fm _ ` a c Standard terminology for this property is that SL2(Q)nSL2(A) is the (projective) limit of the quotients ΓN nSL2(R). To warm up to this situation, replace ΓN and SL2(R) by simpler objects, namely, replace ΓN by NZ and [3] As discussed below, A contains a copy of every Qp as well as R, but is smaller than the product.
Load more

Recommended publications
  • the Group of Homeomorphisms of a Solenoid Which Are Isotopic to The
    . The Group of Homeomorphisms of a Solenoid which are Isotopic to the Identity A DISSERTATION Submitted to the Centro de Investigaci´on en Matem´aticas in partial fulfillment of the requirements for the degree of DOCTOR IN SCIENCES Field of Mathematics by Ferm´ınOmar Reveles Gurrola Advisors Dr. Manuel Cruz L´opez Dr. Xavier G´omez–Mont Avalos´ GUANAJUATO, GUANAJUATO September 2015 2 3 A mi Maestro Cr´uz–L´opez que me ensen´oTodo˜ lo que S´e; a mi Maestra, que me Ensenar´atodo˜ lo que No s´e; a Ti Maestro Cantor, mi Vida, mi Muerte Chiquita. 4 Abstract In this work, we present a detailed study of the connected component of the identity of the group of homeomorphisms of a solenoid Homeo+(S). We treat the universal one–dimensional solenoid S as the universal algebraic covering of the circle S1; that is, as the inverse limit of all the n–fold coverings of S1. Moreover, this solenoid is a foliated space whose leaves are homeomorphic to R and a typical transversal is isomorphic to the completion of the integers Zb. We are mainly interested in the homotopy type of Homeo+(S). Using the theory of cohomology group we calculate its second cohomology groups with integer and real coefficients. In fact, we are able to calculate the associated bounded cohomology groups. That is, we found the Euler class for the universal central extension of Homeo+(S), which is constructed via liftings to the covering space R × Zb of S. We show that this is a bounded cohomology class.
    [Show full text]
  • Class Numbers of CM Algebraic Tori, CM Abelian Varieties and Components of Unitary Shimura Varieties
    CLASS NUMBERS OF CM ALGEBRAIC TORI, CM ABELIAN VARIETIES AND COMPONENTS OF UNITARY SHIMURA VARIETIES JIA-WEI GUO, NAI-HENG SHEU, AND CHIA-FU YU Abstract. We give a formula for the class number of an arbitrary CM algebraic torus over Q. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result. 1. Introduction An algebraic torus T over a number field k is a connected linear algebraic group over k such d that T k k¯ isomorphic to (Gm) k k¯ over the algebraic closure k¯ of k for some integer d 1. The class⊗ number, h(T ), of T is by⊗ definition, the cardinality of T (k) T (A )/U , where A≥ \ k,f T k,f is the finite adele ring of k and UT is the maximal open compact subgroup of T (Ak,f ). Asa natural generalization for the class number of a number field, Takashi Ono [18, 19] studied the class numbers of algebraic tori. Let K/k be a finite extension and let RK/k denote the Weil restriction of scalars form K to k, then we have the following exact sequence of tori defined over k 1 R(1) (G ) R (G ) G 1, −→ K/k m,K −→ K/k m,K −→ m,k −→ where R(1) (G ) is the kernel of the norm map N : R (G ) G .
    [Show full text]
  • Covering Group Theory for Compact Groups 
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 161 (2001) 255–267 www.elsevier.com/locate/jpaa Covering group theory for compact groups Valera Berestovskiia, Conrad Plautb;∗ aDepartment of Mathematics, Omsk State University, Pr. Mira 55A, Omsk 77, 644077, Russia bDepartment of Mathematics, University of Tennessee, Knoxville, TN 37919, USA Received 27 May 1999; received in revised form 27 March 2000 Communicated by A. Carboni Abstract We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism 2 K : G → G in this category. The abelian topological group 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems. c 2001 Elsevier Science B.V. All rights reserved. MSC: Primary: 22C05; 22B05; secondary: 22KXX 1. Introduction and main results In [1] we developed a covering group theory and generalized notion of fundamental group that discards such traditional notions as local arcwise connectivity and local simple connectivity. The theory applies to a large category of “coverable” topological groups that includes, for example, all metrizable, connected, locally connected groups and even some totally disconnected groups. However, for locally compact groups, being coverable is equivalent to being arcwise connected [2]. Therefore, many connected compact groups (e.g. solenoids or the character group of ZN ) are not coverable.
    [Show full text]
  • The Geometry and Fundamental Groups of Solenoid Complements
    November 5, 2018 THE GEOMETRY AND FUNDAMENTAL GROUPS OF SOLENOID COMPLEMENTS GREGORY R. CONNER, MARK MEILSTRUP, AND DUSANˇ REPOVSˇ Abstract. A solenoid is an inverse limit of circles. When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give different groups; in particular, the nicest embeddings are unknotted at each level, and give an Abelian fundamental group, while other embeddings have non-Abelian groups. We show using geometry that every solenoid has uncountably many embeddings with non-homeomorphic complements. In this paper we study 3-manifolds which are complements of solenoids in S3. This theory is a natural extension of the study of knot complements in S3; many of the tools that we use are the same as those used in knot theory and braid theory. We will mainly be concerned with studying the geometry and fundamental groups of 3-manifolds which are solenoid complements. We review basic information about solenoids in section 1. In section 2 we discuss the calculation of the fundamental group of solenoid complements. In section 3 we show that every solenoid has an embedding in S3 so that the complementary 3-manifold has an Abelian fundamental group, which is in fact a subgroup of Q (Theorem 3.5). In section 4 we show that each solenoid has an embedding whose complement has a non-Abelian fundamental group (Theorem 4.3).
    [Show full text]
  • Horocycle Flows for Laminations by Hyperbolic Riemann Surfaces and Hedlund’S Theorem
    JOURNALOF MODERN DYNAMICS doi: 10.3934/jmd.2016.10.113 VOLUME 10, 2016, 113–134 HOROCYCLE FLOWS FOR LAMINATIONS BY HYPERBOLIC RIEMANN SURFACES AND HEDLUND’S THEOREM MATILDE MARTíNEZ, SHIGENORI MATSUMOTO AND ALBERTO VERJOVSKY (Communicated by Federico Rodriguez Hertz) To Étienne Ghys, on the occasion of his 60th birthday ABSTRACT. We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle (Mˆ ,T 1F ) of a compact minimal lamination (M,F ) by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal and examples where this action is not minimal. In the first case, we prove that if F has a leaf which is not simply connected, the horocyle flow is topologically transitive. 1. INTRODUCTION The geodesic and horocycle flows over compact hyperbolic surfaces have been studied in great detail since the pioneering work in the 1930’s by E. Hopf and G. Hedlund. Such flows are particular instances of flows on homogeneous spaces induced by one-parameter subgroups, namely, if G is a Lie group, K a closed subgroup and N a one-parameter subgroup of G, then N acts on the homogeneous space K \G by right multiplication on left cosets. One very important case is when G SL(n,R), K SL(n,Z) and N is a unipo- Æ Æ tent one parameter subgroup of SL(n,R), i.e., all elements of N consist of ma- trices having all eigenvalues equal to one. In this case SL(n,Z)\SL(n,R) is the space of unimodular lattices.
    [Show full text]
  • 3-Manifolds That Admit Knotted Solenoids As Attractors
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 356, Number 11, Pages 4371{4382 S 0002-9947(04)03503-2 Article electronically published on February 27, 2004 3-MANIFOLDS THAT ADMIT KNOTTED SOLENOIDS AS ATTRACTORS BOJU JIANG, YI NI, AND SHICHENG WANG Abstract. Motivated by the study in Morse theory and Smale's work in dy- namics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non- wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds. 1. Introduction The solenoids were first defined in mathematics by Vietoris in 1927 for 2-adic case and by others later in general case, which can be presented either in an abstract way (inverse limit of self-coverings of circles) or in a geometric way (nested intersections of solid tori). The solenoids were introduced into dynamics by Smale as hyperbolic attractors in his celebrated paper [S]. Standard notions in dynamics and in 3-manifold topology will be given in Section 2. The new definitions are the following: Let N = S1 × D2,whereS1 is the unit circle and D2 is the unit disc. Both S1 and D2 admit \linear structures". Let e : N ! N be a \linear", D2-level-preserving embedding such that (a) e(S1 ×∗)isaw-string braid in N for each ∗∈D2,where w>1 in an integer; (b) for each θ 2 S1, the radius of e(θ × D2)is1=w2.
    [Show full text]
  • On the Dynamics of G-Solenoids. Applications to Delone Sets
    Ergod. Th. & Dynam. Sys. (2003), 23, 673–691 c 2003 Cambridge University Press DOI: 10.1017/S0143385702001578 Printed in the United Kingdom On the dynamics of G-solenoids. Applications to Delone sets RICCARDO BENEDETTI† and JEAN-MARC GAMBAUDO‡ † Dipartimento di Matematica, Universita` di Pisa, via F. Buonarroti 2, 56127 Pisa, Italy (e-mail: [email protected]) ‡ Institut de Mathematiques´ de Bourgogne, U.M.R. 5584 du CNRS, Universite´ de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France (e-mail: [email protected]) (Received 2 September 2002 and accepted in revised form 22 January 2003) Abstract.AG-solenoid is a laminated space whose leaves are copies of a single Lie group G and whose transversals are totally disconnected sets. It inherits a G-action and can be considered as a dynamical system. Free Zd -actions on the Cantor set as well as a large class of tiling spaces possess such a structure of G-solenoids. For a large class of Lie groups, we show that a G-solenoid can be seen as a projective limit of branched manifolds modeled on G. This allows us to give a topological description of the transverse invariant measures associated with a G-solenoid in terms of a positive cone in the projective limit of the dim(G)-homology groups of these branched manifolds. In particular, we exhibit a simple criterion implying unique ergodicity. Particular attention is paid to the case when the Lie group G is the group of affine orientation-preserving isometries of the Euclidean space or its subgroup of translations. 1.
    [Show full text]
  • Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves Athanasios Angelakis
    Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves Athanasios Angelakis To cite this version: Athanasios Angelakis. Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves. Group Theory [math.GR]. Université de Bordeaux; Universiteit Leiden (Leyde, Pays-Bas), 2015. English. NNT : 2015BORD0180. tel-01359692 HAL Id: tel-01359692 https://tel.archives-ouvertes.fr/tel-01359692 Submitted on 23 Sep 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Universal Adelic Groups for Imaginary Quadratic Number Fields and Elliptic Curves Proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden op gezag van Rector Magnificus prof. mr. C.J.J.M. Stolker, volgens besluit van het College voor Promoties te verdedigen op woensdag 2 september 2015 klokke 15:00 uur door Athanasios Angelakis geboren te Athene in 1979 Samenstelling van de promotiecommissie: Promotor: Prof. dr. Peter Stevenhagen (Universiteit Leiden) Promotor: Prof. dr. Karim Belabas (Universit´eBordeaux I) Overige leden: Prof.
    [Show full text]
  • LECTURES on HEIGHT ZETA FUNCTIONS of TORIC VARIETIES by Yuri Tschinkel
    S´eminaires & Congr`es 6, 2002, p. 227–247 LECTURES ON HEIGHT ZETA FUNCTIONS OF TORIC VARIETIES by Yuri Tschinkel Abstract.— We explain the main ideas and techniques involved in recent proofs of asymptotics of rational points of bounded height on toric varieties. 1. Introduction Toric varieties are an ideal testing ground for conjectures: their theory is sufficiently rich to reflect general phenomena and sufficiently rigid to allow explicit combinato- rial computations. In these notes I explain a conjecture in arithmetic geometry and describe its proof for toric varieties. Acknowledgments. — Iwould like to thank the organizers of the Summer School for the invitation. The results concerning toric varieties were obtained in collaboration with V. Batyrev. It has been a great pleasure and privilege to work with A. Chambert- Loir, B. Hassett and M. Strauch — Iam very much indebted to them. My research was partially supported by the NSA. 1.1. Counting problems Example 1.1.1.—LetX ⊂ Pn be a smooth hypersurface given as the zero set of a homogeneous form f of degree d (with coefficients in Z). Let N(X, B)=#{x | f(x)=0, max(|xj |) B} n+1 (where x =(x0,...,xn) ∈ Z /(±1) with gcd(xj) = 1) be the number of Q-rational points on X of “height” B. Heuristically, the probability that f represents 0 is about B−d and the number of “events” about Bn+1. Thus we expect that lim N(X, B) ∼ Bn+1−d. B→∞ 2000 Mathematics Subject Classification.—14G05, 11D45, 14M25, 11D57. Key words and phrases.—Rational points, heights, toric varieties, zeta functions.
    [Show full text]
  • Family of Smale-Williams Solenoid Attractors As Orbits of Differential Equations: Exact Solution and Conjugacy
    Family of Smale-Williams Solenoid Attractors as Orbits of Differential Equations: Exact Solution and Conjugacy Yi-Chiuan Chen1 and Wei-Ting Lin2 1)Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwana) 2)Department of Physics, National Taiwan University, Taipei 10617, Taiwanb) (Dated: 25 April 2014) We show that the family of the Smale-Williams solenoid attractors parameterized by its contraction rate can be characterized as a solution of a differential equation. The exact formula describing the attractor can be obtained by solving the differen- tial equation subject to an explicitly given initial condition. Using the formula, we present a simple and explicit proof that the dynamics on the solenoid is topologically conjugate to the shift on the inverse limit space of the expanding map t 7! mt mod 1 for some integer m ≥ 2 and to a suspension over the adding machine. a)Electronic mail: Author to whom correspondence should be addressed. [email protected] b)Electronic mail: [email protected] 1 Fractals are usually described through infinite intersection of sets by means of iterated function systems. Although such descriptions are mathematically elegant, it may still be helpful and desirable to have explicit formulae describing T the fractals. For instance, the famous middle-third Cantor set C = i≥0 Ci, often constructed by the infinite intersection of a sequence of nested sets Ci, where C0 = [0; 1] and each Ci+1 is obtained by removing the middle-third open interval P1 k of every interval in Ci, can be expressed explicitly by C = fx j x = k=1 ak=3 ; ak = 0 or 2 8k 2 Ng.
    [Show full text]
  • Strong Relative Property (T) and Spectral Gap of Random Walks
    isibang/ms/2011/10 November 23rd, 2011 http://www.isibang.ac.in/e statmath/eprints Strong relative property (T ) and spectral gap of random walks C. R. E. Raja Indian Statistical Institute, Bangalore Centre 8th Mile Mysore Road, Bangalore, 560059 India Strong relative property (T ) and spectral gap of random walks C. R. E. Raja Abstract We consider strong relative property (T ) for pairs (¡;G) where ¡ acts on G. If N is a connected nilpotent Lie group and ¡ is a group of automorphisms of N, we choose a ¯nite index subgroup ¡0 of ¡ and obtain that (¡; [¡0;N]) has strong relative property (T ) provided Zariski-closure of ¡ has no compact factor of positive dimension. We apply this to obtain the following: G is a connected Lie group with solvable radical R and a semisimple Levi subgroup S. If Snc denotes the product of noncompact simple factors of S and ST denotes the product of simple factors in Snc that have property (T ), then we show that (¡;R) has strong relative property (T ) for a Zariski-dense closed subgroup ¡ of Snc if and only if R = [Snc;R]. The case when N is a vector group is discussed separately and some interesting results are proved. We also consider actions on solenoids K and proved that if ¡ acts on a solenoid K, then (¡;K) has strong relative property (T ) under certain conditions on ¡. For actions on solenoids we provide some alternatives in terms of amenability and strong relative propertyR (T ). We also provide some applications to the spectral gap of ¼(¹) = ¼(g)d¹(g) where ¼ is a certain unitary representation and ¹ is a probability measure.
    [Show full text]
  • Genus Two Smale-Williams Solenoid Attractors in 3-Manifolds
    Genus two Smale-Williams solenoid attractors in 3-manifolds Jiming Ma and Bin Yu May 3, 2019 Abstract: Using alternating Heegaard diagrams, we construct some 3-manifolds which admit diffeomorphisms such that the non-wandering sets of the diffeomorphisms are composed of Smale- Williams solenoid attractors and repellers, an interesting example is the truncated-cube space. In addition, we prove that if the nonwandering set of the diffeomorphism consists of genus two Smale-Williams solenoids, then the Heegaard genus of the closed manifold is at most two. Keywords:3-manifolds, Smale-Williams solenoid attractors, alternating Heegaard diagram. MR(2000)Subject Classification: 57N10, 58K05, 37E99, 37D45. 1.Introduction For a diffeomorphism of a manifold f : M → M, Smale introduced the notion of hyperbolic structure on the non-wandering set, Ω(f), of f. It is Smale’s long range program to classify a Baire set of these diffeomorphisms, and Ω(f) plays a crucial role in this program. He also arXiv:math/0610474v2 [math.GT] 4 Aug 2009 introduced solenoid into dynamics in [S], in the literature this solenoid is called Smale solenoid or pure solenoid. To carry out this program, Williams defined 1-dimensional solenoid in terms of 1-dimensional branched manifold, which is the generalization of Smale solenoid. There are two methods to define Smale-Williams solenoid: the inverse limit of an expanding map on branched manifold or the nested intersections of handlebodies. Bothe studied the ambient structure of attractors in [B1], through this work, we can see the two definitions above are equivalent. Boju Jiang, Yi Ni and Shicheng Wang studied the global question in [JNW].
    [Show full text]