Additive Polynomial Flows on Tori in Positive Characteristic Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University by Alex Ustian, B.S. Graduate Program in Mathematics The Ohio State University 2012 Dissertation Committee: Nimish Shah, Advisor Vitaly Bergelson Alexander Leibman Copyright by Alex Ustian 2012 Abstract We study one-parameter flows given by additive polynomials on positive characteristic tori from the viewpoint of homogeneous dynamics. We will de- scribe orbit closures, invariant measures, and equidistribution results with re- spect to these flows. We will observe many distinctly positive characteristic phenomena stemming from the existence of flows dependent on the Frobenius map. ii Acknowledgements I would like to thank the Ohio State University for its support during my graduate studies, and I would like to thank my advisor Nimish Shah. His patient listening and many helpful suggestions, clarifications, and encouragement have been invaluable. I would like to thank Vitaly Bergelson and Jim Cogdell, who over the years have taught me many things that have broadened my mathematical horizons. This project was started under the recommendation of Manfred Einsiedler while the author was visiting the ETH. I would like to thank him for his guidance during the initial stages of this paper and for all the knowledge he has shared with me throughout my graduate studies. I would also like to thank the ETH for hosting me and providing a great atmosphere to work on mathematics. I would like to thank my parents Jim and Claudia, and my sister Janelle. Their unwavering support of me and my endeavors has given me the confidence to always pursue my interests. Finally, I would like to thank my wife Stacey. For sharing the highs and the lows of this process with me and giving me a never-ending source of motivation to continue onward. iii Vita June 2000.............................................Sylvania Northview High School June 2005.............................................B.S. Mathematics, The Ohio State University June 2009 - June 2010..........................Graduate Teaching Associate, Department of Mathematics, ETH Z¨urich June 2005 to present............................Graduate Teaching Associate, Department of Mathematics, The Ohio State University Fields of Study Major Field: Mathematics iv Table of Contents Abstract . ii Acknowledgements . iii Vita . iv 1 Introduction 1 1.1 Background . 1 1.2 Positive characteristic tori . 2 1.2.1 What are \Z" and \R"...................... 2 1.2.2 The positive characteristic torus . 5 1.3 Preliminary discussion and motivation . 6 1.3.1 Continuous one-parameter additive actions . 6 1.3.2 Ratner's theorems . 8 1.3.3 Initial contrast of positive versus zero characteristic . 10 1.3.4 Two examples . 10 1.3.5 Hausdorff dimension of a measure . 11 2 Setup 14 2.1 Decompositions . 14 2.1.1 Decomposing K over K(pn)................... 15 2.1.2 Self-similarity of K ........................ 16 2.1.3 Useful lemmas involving ρpn ................... 17 2.1.4 Example . 18 2.2 Auxiliary one parameter additive flows . 18 v 2.2.1 Classical flows . 18 2.2.2 Separated flows . 19 2.2.3 Separated and classical flows associated to a general flow . 19 2.2.4 One parameter rational Frobenius subgroups . 20 3 Statement of main results 21 3.1 Classical linear case . 21 3.2 Separated case . 22 3.3 General case (single additive polynomial action) . 23 3.4 Geometry of F (Uq)............................ 24 3.5 Towards conjecture 3.4.1 . 25 3.6 Known cases of conjecture 3.4.1 . 26 3.7 Applications . 28 3.7.1 Hausdorff dimension . 28 3.7.2 Unique ergodicity . 29 3.7.3 Applications to unipotent flows of SL2 . 30 3.7.4 A Diophantine application . 32 4 Proofs of theorems 3.1.1 - 3.3.6 34 4.1 Proofs of the classical case . 34 4.2 Proofs of the separated case . 36 4.3 Proofs of the general case (single additive polynomial action) . 38 5 Preliminaries to theorem 3.5.1 40 5.1 Finding G(Uq) and introducing linear terms . 42 5.2 Description of Θ . 45 5.3 Description of dΘ............................. 46 vi 5.3.1 Relating the βjmiγ(j)n to restriction of scalars . 47 5.3.2 Contribution from a single polynomial . 49 5.4 Algebraic geometry and analytic manifold preliminaries . 54 5.4.1 Regularity . 55 5.4.2 Tangent spaces . 55 5.4.3 Analytic versus Zariski tangent spaces aside . 56 5.4.4 Immersions, submersions, and subimmersions . 57 5.4.5 Injective immersions and immersed submanifolds . 57 6 Proof of theorem 3.5.1 59 6.1 General case with regular assumption (proof of theorem 3.5.1) . 59 6.2 Remarks on the regularity of ker Θ . 65 6.2.1 Separability of Θ . 66 6.2.2 Another interpretation of the possible failure of ker Θ to be a manifold . 68 7 Proofs of the embedded submanifold theorems (proofs of theorems 3.6.1, 3.6.5, 3.6.3, and 3.6.2) 72 7.1 Closed orbit case and corollary 3.6.5 . 72 7.2 Monomial case . 74 7.3 Separated case . 74 7.3.1 Describing Π . 75 7.3.2 Describing G(Us¯) and Θ . 77 7.3.3 Computing rank dΘ ....................... 81 8 Proofs of the applications 82 8.1 Hausdorff dimension . 82 vii 8.2 Unique ergodicity . 84 8.3 Special unipotents in SL2 ........................ 85 8.4 Diophantine application . 87 9 References 91 viii 1 Introduction 1.1 Background In this paper we would like to continue the function field/number theory analogy to the ergodic theory of flows on homogeneous spaces. Much of the development of homogeneous dynamics in positive characteristic has mirrored the historical devel- opment and naturally builds off of the characteristic zero theory. For example, Mo- hammadi has studied a positive characteristic analogue of Oppenheim's conjecture [15] and the case of actions by horospherical groups, [14] and Einsiedler and Moham- madi have studied joinings of actions given by maximal horospherical groups [4]. In Einsiedler and Ghosh [3], a characteristic zero argument developed by Einsiedler in [2] for actions by semisimple groups was modified to work in positive characteristic with restrictions on the characteristic. However, a full version of Ratner's measure classification theorem in positive characteristic has not yet been achieved. In characteristic zero the cases of actions by continuous one parameter subgroups on tori have proven to be useful guides towards understanding unipotent actions on more sophisticated groups such as SLn. We will explain how to construct the analogue of the torus R=Z in positive characteristic, examine the one parameter subgroups, describe the orbit closures (subgroups) and invariant measures of actions coming from these one parameter subgroups, and see many contrasts to the characteristic zero case. For example, non-algebraic subgroups and fractal behavior will arise during our discussion and are needed to help describe the orbit closure subgroups. 1 1.2 Positive characteristic tori 1.2.1 What are \Z" and \R" Let K be a field, throughout this paper, unless otherwise noted, p is the character- istic of K. As the familiar setup for ergodic theory on homogeneous spaces is in characteristic zero, we have included an introductory section explaining the origin and construction of our positive characteristic torus. From a number theoretic standpoint, we obtain the characteristic p torus by mim- icking the construction of the regular torus, R=Z. In characteristic zero, we first start off with our base ring Z and form its field of fractions Q. We complete it with respect to the Archimedean metric to obtain R and form the quotient to get R=Z. Now in characteristic p, Fp[t] is still a Dedekind domain and has all of the associ- ated \number theory" attached to it. (i.e. ideal/valuation theory) So from Fp[t] we form the field of fractions Fp(t) and look at the available valuations. Let R = Z or Fp[t], and K = Q or Fp(t), then one uses the standard construction to get valuations (hence norms) on K corresponding to prime ideals in R. More specifically, using the unique factorization of R-fractional ideals in K, for a prime ideal p ⊂ R and a 2 K there exists an integer νp(a) 2 Z such that (a) = pνp(a)q (1.2.1) for some R-fractional ideal q ⊂ K relatively prime to p. From here one may pick any 0 < c < 1 and define equivalent norms νp(a) jajp = c : Let Kp denote the completion of K with respect to the norm j · jp. Notice that 2 R ⊂ fa 2 Kp j jajp ≤ 1g = Rp and p ⊂ fa 2 Kp j jajp < 1g = pp. It may then be shown that for such a setup, R=p ' Rp=pp; and that the elements of Kp have the form of a Laurent series, as a set Kp = (R=p)((π)), where π generates the principal ideal pp,(π) = pp. As in the characteristic zero case, there is indeed an additional metric not coming from a prime ideal of Fp[t] on Fp(t). The remaining valuation is defined as for f=g 2 Fp(t), f ν = deg g − deg f; 1 g i.e. the degree valuation. Place at 1 However unlike characteristic zero, this additional \place at 1" can be rigged to still have the same general form as the above prime ideal valuation construction. For 1 f=g 2 Fp(t) we may factor out powers of t to see f=g = (1=t)kq; (1.2.2) −1 where k 2 Z and q is a power series in 1=t (i.e.