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On the Complexity of Anosov Saddle Transitions✩ Michael Maller Mathematics Department, Queens College, CUNY, Flushing, NY 11367, USA Article Info a B S T R a C T

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On the Complexity of Anosov Saddle Transitions✩ Michael Maller Mathematics Department, Queens College, CUNY, Flushing, NY 11367, USA Article Info a B S T R a C T Journal of Complexity ( ) – Contents lists available at ScienceDirect Journal of Complexity journal homepage: www.elsevier.com/locate/jco On the complexity of Anosov saddle transitionsI Michael Maller Mathematics Department, Queens College, CUNY, Flushing, NY 11367, USA article info a b s t r a c t Article history: We study the computational complexity of some problems which Received 25 September 2014 arise in the dynamics of certain hyperbolic toral automorphisms, Accepted 18 March 2017 using the shadowing lemma. Given p; q 2 T2, saddles of common Available online xxxx period n, and rational r > 0 sufficiently small, we study how quickly orbits .f n/N .z/ transit from a basin of p, the open ball Keywords: B.p; r/ to one of q, B.q; r/. A priori, z is of arbitrary precision. We Anosov diffeomorphism compute a finite precision L, polynomial in the size of an instance, Shadowing n N Computational complexity so that if an orbit .f / .z/ makes such a transition, so does a reliable L-precision pseudo-orbit, but at the price of possibly using duration N C 1. If we allow duration N C 2, this precision depends on n and r but is independent of N. It follows that in an appropriate model of computation such a saddle transition problem is in Oracle NP i.e. in NP up to a restricted oracle, and we show there exist closely related problems which are in NP. ' 2017 Elsevier Inc. All rights reserved. 1. Introduction We review some necessary notions from dynamical systems; in particular we need to compute estimates for some of the constants. Except for N and n, which we encode in unary, the size of inputs is measured in the bit model, and the cost of computation by Turing Machine operations. Thus we consider complexity notions for problems defined over the real numbers in the Turing machine model. A general reference in this area is [4]. Our approach was motivated by the Blum–Shub–Smale model for machines over the real numbers R [1,2]. It is a pleasure to acknowledge very helpful conversations with Michael Shub. The author is grateful to the referee for many important suggestions and improvements. I Communicated by P. Hertling. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jco.2017.03.002 0885-064X/' 2017 Elsevier Inc. All rights reserved. 2 M. Maller / Journal of Complexity ( ) – Let A be a .2 × 2/ hyperbolic non-negative integer matrix, det.A/ D ±1, with eigenvalues 2 0 < kλk < 1 < kµk, and eigenbasis vµ; vλ. A induces a diffeomorphism f D fA of the 2-torus T . 2 A 2 R −−−−! R ? ? ?π ?π: y y f T2 −−−−!A T2 These hyperbolic toral automorphisms (HTA) are the paradigm examples of Anosov diffeomor- phisms f V M ! M, which are hyperbolic on all of M. The periodic points of fA are the points with both 2 2 coordinates rational, Q /π, so the periodic points of fA are dense in T . If p is a periodic point of period n, writing g D f n, the derivative Dg D Df n D An so by Hartman's Theorem g is locally conjugate to An. n −1 Hence there is a radius, r ≤ kλk =4 D rn suffices, so that B.p; r/ [ g.B.p; r// [ g .B.p; r// lies in the C domain of the local conjugacy [3]. Let p, q be two such points of common period n, r 2 Q ; 0 < r ≤ rn. We define a saddle transition problem: Saddle transition problem ST .HTA/. C n Instance: .A; p; q; n; r; N/; N 2 Z ; p 6D q, g D .fA/ . Does there exist rational z 2 B.p; r/, so gN .z/ 2 B.q; r/? In this paper we will study the special case where the matrix A is symmetric, and write this restricted problem as ST .SHTA/. Observe that if a real solution z exists, so does a rational one. We will abbreviate .A; p; q; n/ D α. Of course saddle transition problems can be defined more generally, but to be computationally meaningful f ; p; q must all be effectively encoded as inputs [8,5]. The classic example of an HTA is 2 1 A D 0 1 1 p p − C with eigenvalues λ D 3 5 ≈ 0:38197 < 1 < µ D 3 5 ≈ 2:618. For an HTA, A, with det A DC1 0 2 p 0p 2 2 2 D Tr.A/− Tr.A/ −4 D Tr.A/C Tr.A/ −4 ≤ ≤ and eigenvalues λ 2 ; µ 2 observe 0 < λ λ0 < 1 < µ0 µ, n n n n Tr.A/ ≥ Tr.A0/ D 3 and Tr.A / D µ C λ D dµ e. Observe that A0 is symmetric, so the eigenvectors vλ corresponding to λ0 and vµ corresponding to µ0 are orthogonal. Henceforth we will write f for fA, and for technical reasons we will restrict to det.A/ DC1. 2 2 2 It is instructive to consider A0 V R ! R as a dynamical system. The origin θ 2 R is a hyperbolic s n fixed point. Along the eigenspace E generated by vλ orbits fA .z/jn 2 Zg contract toward θ by λ0, on u that generated by vµ, E , orbits expand away from θ by µ0, all other orbits are hyperbolic, asymptotic to Es in the past and Eu in the future. The projections W s(π(θ// D π.Es/ and W u(π(θ// D π.Eu/ wind densely about T2. If p is a periodic point of f of period n since Df n D An the same local picture is repeated at p. We can define local and global stable and unstable manifolds for all points in T2 s 2 n n n n Wϵ .x/ D fy 2 T jd.f .x/; f .y// ≤ ϵ; n ≥ 0 ^ d.f .x/; f .y// ! 0; n ! C1g s [ −n s n W .x/ D f .Wϵ .f .x///; n≥0 u 2 n n n n Wϵ .x/ D fy 2 T jd.f .x/; f .y// ≤ ϵ; n ≤ 0 ^ d.f .x/; f .y// ! 0; n ! −∞} u [ n u −n W .x/ D f .Wϵ .f .x///: n≥0 s u 2 For fA, W .x/ and W .x/ are simply the π images of the lines in R through x parallel to the 2 2 s u contracting and expanding eigenspaces of A V R ! R , while Wϵ .x/ and Wϵ .x/ are π images of the 2 line segments in R with center x and length 2ϵ parallel to the contracting and expanding eigenspaces 2 2 of A V R ! R . s u 2 2 2 Each W .p/, W .p/ is a 1–1 immersion winding densely about T . The dynamics of A0 V R ! R is simple, and reproduced locally at periodic points. However observe that for periodic points p and q .
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