Hyperbolic Dynamical Systems

Transcription

1 Hyperbolic Dynamical Systems Mark Pollicott Warwick University 27 June, / 35

2 An overview In this lecture I would like discuss the following three themes: 1 Geodesic flows and their generalizations (Anosov flows, Axiom A flows, etc.); 2 Closed geodesics and zeta functions for compact surfaces; 3 Closed geodesics and zeta functions for infinite area surfaces The zeros of the zeta function for compact surface have a spectral interpretation. The zeros of the zeta function for infinite area surfaces are more mysterious. 2 / 35

3 A tale of two cities: schools of Dynamical Systems in the 1960s It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness... In the 1960s there were two major schools working on related types of dynamical systems. Moscow. There was important work by Anosov, Sinai and others generalizing dynamical properties of geodesic flows on negatively curved manifolds (i.e., Anosov flows). Berkeley. There was seminal work by Smale, which influenced Anosov, and his student Bowen, which extending properties of Anosov flows to more general Axiom A flows. These two developments were closely related... 3 / 35

4 The chronology In 1960, on the beaches of Rio, Smale found an example of an open set of diffeomorphisms of the sphere with infinitely many periodic points ( horseshoes ) The diffeomorphism G has an invariant set consisting of all points x which remain in H 0 H 1, i.e., G n (x) H 0 H 1, for all n H. Moreover, each infinite repeating sequence H i0, H i1,, H in 1 H i0, H i1,, H in 1 H i0, H i1, gives rise to a periodic orbit. The bad news (for Smale) was that this example provided a counter-example to his own conjecture, to the effect that typically there were only finitely many periodic orbits. The good news (for Anosov) was that when Smale subsequently gave a talk on this result in Kiev, Anosov read the abstract and was motivated to consider analogous problems for geodesic flows (and their generalizations). 4 / 35

5 Geodesic flows on negatively curved manifolds γ Consider the geodesic flow on a negatively curved surface. x v x v Definition (Geodesic flow) Let V be a closed surface with negative curvature. Let M = {(x, v) TV : v x = 1} be the three dimensional space of tangent vectors of unit length ( directions ). Let φ t : M M be the geodesic flow, i.e., φ t(v) = γ(t) where γ : R V is the unit speed geodesic with γ(0) = v. Periodic orbits τ of period λ(τ) correspond to closed geodesics of length l(γ). 5 / 35

6 Anosov flows The geodesic flow on a negatively curved surface is a special case of an Anosov flow, i.e., a flow which transverse to the flow direction is uniformly hyperbolic, (i.e., it stretches in one direction E u and it contracts in another direction E s.) s E x E u φ t x Figure: (a) The hyperbolicity transverse to the orbit of an Anosov flow; (b) D.V. Anosov. Let τ be a closed orbit and let λ(τ) > 0 be its period ( i.e., φ λ(τ) (ξ) = ξ for ξ τ). We say that the Anosov flow is (topologically) weak mixing if the periods are not all integer multiples of a fixed constant. Theorem The geodesic flow on a (variable) negatively curved compact surface is a weak mixing Anosov flow. 6 / 35

7 Axiom A flows Smale generalized the general idea of Axiom A flows. Figure: The definiton from Smale s influential 1967 survey Differentiable Dynamical Systems published in the Bull. Amer. Math. Soc. In fact, Axiom A was original called Axiom A but the prime seems to have been dropped over the years... nor did the other property Axiom B ever catch on... 7 / 35

8 Example: Geodesic flows on infinite area surfaces We can consider a surface of infinite volume (without cusps) such as a pair of pants. Although the unit tangent bundle of the surface is infinite the recurrent set Ω (i.e., an invariant set for the geodesic flow) is compact. Transverse to the flow direction it has a cross-section which is a Cantor set. Since it is difficult to draw pictures of infinite area surfaces it is often convenient to cut off the (three) infinite funnels by closed geodesics. The recurrent set Ω is contained within the unit tangent bundle of the resulting surface with boundary. 8 / 35

9 Three types of zeta functions 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces Having introduced the geodesic flow (on both compact surfaces and infinite area surfaces) we want to introduce associated zeta functions. We will consider three types of zeta function. 1 The Riemann zeta function for prime numbers (number theory - motivation only); 2 The Selberg zeta function for lengths of closed geodesics for compact surfaces without boundary of constant negative curvature (for historical background); and 3 The Selberg zeta function for lengths of closed geodesics for compact surfaces with boundary of constant negative curvature (the newer material). Aim are complex functions and we want to understand properties of their domains. 9 / 35

10 The Riemann zeta function 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces The Riemann zeta function is the complex function (introduced by Euler) ζ(s) = which converges for Re(s) > 1. However, it is convenient for us to write this in the equivalent form as an Euler product n=1 1 n s ζ(s) = p ( 1 p s ) 1 where the product is over all primes p = 2, 3, 5, 7, 11,. Question What are the properties of ζ(s)? 10 / 35

11 Properties of the Riemann zeta function 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces The Riemann zeta function is analytic and non-zero for Re(s) > 1. Analytic extension Analytic Pole Moreover, 1 ζ(s) has a simple pole at s = 1; 2 ζ(s) has an analytic extension to C {1}. The following famous conjecture was formulated by Riemann in 1859 (repeated as Hilbert s 8th problem at the ICM-1900, and it is also one of the Million Dollar Millennium Problems from the Clay institute.). Riemann Hypothesis The zeros in Re(s) > 0 lie only on the line Re(s) = / 35

12 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces Riemann Hypothesis The conjecture appears in Riemann s only work on number theory, in his report to the Berlin Academy of Sciences upon his election in 1859: Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of my investigation 12 / 35

13 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces The Hilbert-Polya approach to the Riemann hypothesis Hilbert and Polya are associated with the idea of tying to understand the location of the zeros in terms of eigenvalues of some (as of yet) undiscovered self-adjoint operator (which necessarily has real eigenvalues somehow related to the zeros). This idea has yet to reach fruition for the Riemann zeta function (despite interesting work of Connes, etc.) but the approach works particularly well for the geometric analogue... the Selberg Zeta function 13 / 35

14 2. Selberg Zeta Function 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces Geometric Analogy We would like to replace prime numbers by closed geodesics on manifolds. To begin, assume that V is a compact surface with constant curvature κ = 1. We denote by γ one of the countably many closed geodesics on V (there is exactly one in every conjugacy class of the fundamental group and their lengths tend to infinity). V V V 0 Τ 0 Τ 0 Τ 0 Τ Problem How can we associate a zeta function to this countable set of real numbers? 14 / 35

15 The Selberg Zeta function 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces Given a closed geodesic γ we again denote its length by l(γ). Definition The Selberg Zeta function Z(s) = (1 e (s+n)l(γ)) n=0 γ for R(s) > 1. Remark. It might seem more natural to define the zeta function by γ ( 1 e ssl(γ)) 1 for R(s) > 1, by analogy with the Riemann zeta function ζ(s) = ( τ 1 e sλ(τ) ) 1. But Z(s) and ζ(s) are easily related by ζ(s) = Z(s+1) and have the same zeros/poles in Re(s) > 0. Z(s) 15 / 35

16 Properties of the Selberg zeta function 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces With this definition we have that: ζ(s) has a simple zero at s = 1; ζ(s) otherwise has no zeros on Re(s) = 1; and Theorem (Selberg (1956)) ζ(s) extends analytically to the entire complex plane C. Problem How are the zeros of Z(s) in 0 < Re(s) < 1 related to an operator? Traditionally, one uses the Selberg Trace Formula to prove the theorem and locate the zeros, using the Laplacian. More precisely 16 / 35

17 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces Eigenvalues of the Laplacian and zeros of Z(s) for compact surfacess Let ψ n + λ nψ n = 0, where : C (V ) C (V ) is the self-adjoint laplacian. There are countably infinity of real eigenvalues 0 = λ 0 < λ 1 λ 2 +. Lemma (Weyl s Theorem (1911)) The eigenvalues satisfy Card{λ n T } Area(V ) T as T +. 4π Theorem The zeros of Z(s) in the strip 0 < Re(s) < 1 are of the form s n = lie on {σ : 0 < σ < 1} { 1 + it : t R}. 2 1 λn and thus 4 17 / 35

18 Partial analogue of the Riemann Hypothesis 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces This characterization of the zeros of Z(s) by the spectrum of the Laplacian on V leads to a weak analogue of the Riemann Hypothesis. Corollary (Analogue of the Riemann Hypothesis, Selberg (1956)) There exists ɛ > 0 such that the Selberg zeta function Z(s) has a non-zero analytic extension to Re(s) > (1 ɛ) except for the simple zero at s = ɛ Problem What happens if we allow the surface to have a boundary? The definition of the zeta function is the same but the positions of the zeros are somewhat different. 18 / 35

19 Surfaces with boundary 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces Let us consider a very specific example of a compact surface with boundaries. 2l 1 2l 2 2l 3 Topologically pair of pants V is a 3-punctured sphere; Assume it has a metric of constant negative curvature 1 and boundary curves which are geodesics; the Riemann metric is uniquely defined by the lengths of the three boundary geodesics: V = V (l 1, l 2, l 3 ) ; It again possess countably many of closed geodesics {γ n} of the lengths 0 < l(γ 1 ) < l(γ 2 ) <... < l(γ n)... By a symmetric pair of pants we mean that l 1 = l 2 = l 3 =: b. We will henceforth restrict to symmetric pairs of pants. The terminology is due to W. Thurston and is decidedly american). 19 / 35

20 The Selberg zeta function Z(s) again 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces Given a closed geodesic γ we again denote its length by l(γ). Definition We can (again) define a Selberg Zeta function ( Z(s) = 1 e (s+n)l(γ)) for R(s) > 1. n=0 γ The following properties are easy to establish. Question There exists 0 < d < 1 such that the zeta function Z(s) is analytic for Re(s) > d. Z(s) has a simple zero at s = d. Does Z(s) still have an analytic extension to C? Where are the zeros? 20 / 35

21 Analytic extension 1. The Riemann Zeta Function 2.Selberg Zeta function 3. for infinite area surfaces We begin with the question of the analytic extension. 0 d Fortunately, we have the analogue of the result for surfaces without boundary: Theorem (Grothendieck-Ruelle) The zeta function Z(s) has an analytic extension to C. However, the method of proof is very different to that of Selberg, and entirely dynamical (e.g., markov sections, symbolic dynamics, transfer operators, etc.) 21 / 35

22 Borthwick and his zeros Borthwick zeros Scaling the pictures as b + A couple of years ago I heard a talk in Roscoff by David Borthwick, on empirical plots of zeros of the zeta function for the geodesic flow on a pair of pants. Roscoff is perhaps better known for its onions, which were highly praised by Alexander Dumas, author of The three musketeers and The Count of Monte Cristo, etc., in his dictionary of cooking. 22 / 35

23 The zeros for Z(s) Borthwick zeros Scaling the pictures as b + Question Where the zeros for Z(s) in 0 < Re(s) < 1? Plots of the zeros were made by David Borthwick, for example b = 3π = , say. (The value of b has to be relatively large to get accurate numerical results) ,3, Here the scaling in the vertical axis gives the impression of a continuous curve, but there are just lots of isolated zeros which seem close together in the vertical direction. Clearly, this is very different from the surfaces without boundary where the zeros lie on the vertical line Re(s) = / 35

24 Empirical obervations Borthwick zeros Scaling the pictures as b + There are some observations which it would be nice to have (at least) an heuristic explanation. 1 The apparent vertical aperiodicity of the pattern of zeros is approximately πe b. 2 The pattern of zeros appears to lie on a finite number of curves. Remark. There is a nice result by Tobias Weich (CMP, 2015) which describes how the asymptotic spacing between zeros tends to zero as b / 35

25 The apparent (but not real) periodicity of Z(s) Borthwick zeros Scaling the pictures as b + In fact, the picture cannot be really periodic in the vertical direction (otherwise there would be infinitely many zeros above s = d which it is relatively easy to show isn t the case). Figure: A more careful look at the apparent periodicity (which isn t actually periodic) 25 / 35

26 Aside: Previous results and conjectures Borthwick zeros Scaling the pictures as b + There are much stronger results, more in the spirit of the Riemann Hypothesis. Theorem (F. Naud) If d is the largest zero on the real axis then there exists ɛ > 0 such that all the non-real zeros are in the region Re(s) < d ɛ. There is a related result which replaces ɛ > 0 by a definite value, at the expense of a slightly weaker statement. Theorem (D.Jakobson-F. Naud) There are only finitely many zeros with Re(s) > d(1 2d) 2. There is a conjecture due to Jakobson-Naud that there are only finitely many zeros in Re(s) > d + ɛ for any ɛ > / 35

27 The pattern of the zeros Borthwick zeros Scaling the pictures as b + As b + the regions we are interested in get taller and narrower. So to compare them properly let us affinely transform the portion of the vertical strip [0, d] i[0, πe b /2] = {s = σ + it : 0 < σ < d and 0 < t < πe b /2} depending on b + to a standard rectangle [0, 1] [0, π/2] = {s = σ + it : 0 < σ < 1 and 0 < t < π/2} by the map A b : σ + it σ + it = (bσ) + i(te b ). πe b /2 A b π/2 1 d 27 / 35

28 Convergence of rescaled zeros Borthwick zeros Scaling the pictures as b + Let us consider the images of the zeros for Z(s) = Z(s, b) under the map A b : [0, d] i[0, πe b /2] [0, 1] i[0, π/2]. Aim We want to show that as b + these (rescaled) zeros converge on a finite family of very simple curves. But then this prompts the following question. Question But what can these limiting patterns of curves be? 28 / 35

29 Curves of Zeros Borthwick zeros Scaling the pictures as b + These limiting curves in [0, 1] i[0, π/2] are explicitly given by the formulae { } C 1 = ln e 2it it t R ; { } C 2 = ln e 2it 1 + it t R ; { 1 2 C 3 = 2 ln e 4it e 2it } 4 3e 4it ln it t R ; { 1 2 C 4 = 2 ln e 4it + e 2it } 4 3e 4it ln it t R. Theorem (P.-Vytnova) The rescaled zeros Z Vb (s) (i.e., the zeros of σ + it Z(σ/b + ie b t, b)) converge on C 1 C 2 C 3 C 4. More precisely, for any ɛ > 0 there exists b 0 > 0 such that for any b > b 0 : {σ + it : Z(σ/b + ie b t) = 0} B( 4 i=1 C i, ɛ) 29 / 35

30 The four curves Borthwick zeros Scaling the pictures as b + Using Mathematica we can plot the four curves: C 1, C 2, C 3, C 4 and their union 4 i=1 C i We can compare these limiting plots with Borthwick s original experimental plot. 30 / 35

31 The source of the four curves Borthwick zeros Scaling the pictures as b + The proof is based on approximating the (affinely rescaled) zeta function Z(σ/b + ite b, b) by det(i wb(e 2it )) with w = w(σ, t) := e σ+iteb and 1 z 0 0 z 2 z z 1 z 2 z z z z B(z) = 2 z 2 z z z z 2 1 z z z z 1 We can then compare affinely scaled Selberg zeta function: Theorem (P.- Vytnova) For any 0 < k < 2 as b, ( σ ( ) sup Z X 0<σ<d, t e kb b + iteb) det(i e σ+iteb B(e 2it 1 )) = O b. The four curves C 1, C 2, C 3, C 4 then arise naturally from the eigenvalues of B(e 2it ). The scale t e kb is bigger than the apparent period πe b. 31 / 35

32 Bowen-Series coding Borthwick zeros Scaling the pictures as b + Question Where does the matrix come from? If we set z = 1 then we get a matrix with entries 0 and 1: B(1) = This is an example of a transition matrix used in coding geodesics using spaces of sequences. The fundamental group of the pair of pants is a free group (generated by three reflections R 1, R 2, R 3 in the Poincaré disk). Periodic geodesics correspond to repeating strings R i0, R i1, R i2n 1, R i0, R i1, which are reduced, i.e., no reflection follows itself, i.e., R is R is+1 The matrix has rows and columns indexed by R i R j with i j and encodes this information: the entry 1 appears when the row R i R j and the column R i R j satisfy R j R i. 32 / 35

33 Bowen-Series coding Borthwick zeros Scaling the pictures as b + The basic coding of geodesics dates back to the work of Koebe (1927) and Morse (1921). The method was useful to prove ergodic properties of the geodesic flow in the work of Artin and Hedlund. Bowen and Series considered the boundary coding of geodesics (1980) in a posthumous paper. Adler and Flatto has a slightly different viewpoint (1984). A more modern approach came from the work of Cannon (and is nicely described in the book of de la Harpe and Ghys). 33 / 35

34 Bowen-Series coding Borthwick zeros Scaling the pictures as b + Question How does the coding give us the zeta function? The dynamics is coded by infinite sequences and the transition matrix B(1) containing 0s and 1s. The periodic sequences correspond to closed geodesics on V. The powers of z keep track of how successive geodesic arcs cross the surface are related (which in turn modifies the first guess for the length of kb (k N 0 by adding suitable le b (l N 0 )). 34 / 35

35 Lengths of Closed Geodesics Borthwick zeros Scaling the pictures as b + R 2 R 1 γ 131 R 3 γ 132 R 2 R 1 Consider a closed geodesic γ on X (b) corresponding to a cutting sequence of period 2n j 2n 1 j 2n j 2n+1, where j k {1, 2, 3}, j k j k+1 for 1 k 2n and j 2n j 1. Then where l(γ) = 2nb + c(γ)e b + O(e 2b ), c(γ) = #{1 k 2n : j k j k+2 mod 2n }. 35 / 35