DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2019 On the dynamics of a family of critical circle endomorphisms NILS HEMMINGSSON KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES On the dynamics of a family of critical circle endomorphisms NILS HEMMINGSSON Degree Projects in Mathematics (30 ECTS credits) Degree Programme in Mathematics (120 credits) KTH Royal Institute of Technology year 2019 Supervisor at KTH: Kristian Bjerklöv Examiner at KTH: Kristian Bjerklöv TRITA-SCI-GRU 2019:353 MAT-E 2019:83 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci On the dynamics of a family of critical circle endomorphisms Nils Hemmingsson 2019 Abstract In this thesis we study two seperate yet related three parameter-families of continuously differentiable maps from the unit circle to unit circle which have a single critical point. For one of the families we show that there is a set of positive measure of parameters such that there is a set of positive measure for which all points in the latter set, the derivative experiences exponential growth. We do so by applying a similar methodology to what Michael Benedicks and Lennart Carleson used to study the quadratic family. For the other family we attempt to show a similar but weaker result using a similar method, but do not manage to do so. We expound on what difficulties the latter family provides and what features Benedicks and Carleson used for the quadratic family that we do not have available. 1 Om dynamiken hos en familj kritiska cirkel-endomorfier Sammanfattning I den h¨aruppsatsen studerar vi tv˚aolika men relaterade treparameterfamiljer av kontinuerligt differentierbara avbildningar fr˚anenhetscirkeln till enhetscirkeln som har exakt en kritisk punkt. For den ena familjen visar vi att det finns en m¨angdav positivt m˚attav parametrar s˚adanaatt det finns en m¨angdav pos- itivt m˚atts˚aatt f¨orvarje punkt i den senaren¨amndem¨angdenerfar derivatan exponentiell tillv¨axt. Vi uppn˚ardetta genom att anv¨andaen metod som lik- nar den som Michael Benedicks och Lennart Carleson anv¨andef¨oratt studera den kvadratiska familjen. F¨orden andra familjen f¨ors¨oker vi visa ett liknande men svagare resultat genom att anv¨andaen liknande metodik men misslyckas. Vi diskuterar och f¨orklararvilka sv˚arigheterden senare familjen ger och vilka egenskaper som Benedicks och Carleson anv¨andersig av hos den kvadratiska familjen som v˚arfamilj saknar. 1 Acknowledgements: I would like to thank my supervisor Kristian Bjerkl¨ov for very helpful advice and supporting enthusiasm throughout the work with this thesis. 2 Contents 1 Introduction 4 2 Hyperbolicity and Structural Stability 4 3 Lyapunov exponents 6 4 One-dimensional maps 7 5 The chosen model and main results 10 5.1 Results . 12 6 Preliminaries 12 6.1 Definitions . 12 6.2 Defining how small 1 will be . 14 6.3 Estimates on our newly defined numbers . 14 7 Analysis 16 7.1 Idea of the construction . 16 7.2 Finding an estimate for the derivative at times Ti ....... 16 7.3 The derivative grows exponentially . 19 8 There is a set of positive measure for which A1 is fulfilled 21 9 Analyzing more than the critical point 25 10 There are a such that the Lyapunov exponent of the critical point is negative 27 11 Back to analyzing g 28 11.1 The proof of Benedicks and Carleson . 30 11.1.1 Proving the induction assumptions . 37 11.2 Difficulties with g ......................... 40 12 Future work 42 12.1 Positive Lyapunov exponent . 42 12.2 Negative Lyapunov exponent . 43 12.3 Sub-exponential growth for g .................. 46 Bibliography 46 3 1 Introduction Given the simplicity of many questions posed in the area of dynamical sys- tems, it might come as a surprise that it is a relatively new subject. In the late nineteenth and early twentieth century, Henri Poincare published two books on Celestial Mechanics which many view as the birth of the theory of dynamical systems [13]. Suppose that we have a function f : M ! M where M is a manifold. The double (f; M) is called a dynamical system and the main objective of the theory is to analyze what happens when f is iterated multiple times for certain x. This is possible since for every x 2 M, f(x) 2 M and by induction it follows that xn+1 = f(xn) 2 M; x0 = x. We n write xn as f (x). Beyond that the theory is beautiful, the study of dynam- ical systems is interesting because they appear in many different sciences such as physics, chemistry and biology. Possible questions that may arise is whether the orbit, which is the set fxn : n 2 Ng (or fxn : n 2 Zg in the case that f is invertible), is dense, if the orbit is attracted to some particular cycle and so forth. The existence of periodic points, which occur if there is an n > 0 such that xn = x0, is also of interest. These questions have been of great interest during the past century and many problems have thereby been solved. However, to answer whether some dynamical system has periodic points oftentimes turn out to be substantially more difficult than what the simplicity of posing the question might suggest. In a similar manner, proofs of conjectures which are easy to state and appear to be true have eluded mathematicians for a long time. The systems explained above are what we call discrete dynamical systems. There are also continuous ditto, but as this thesis focus on discrete systems we will forgo the definition of the continuous case. 2 Hyperbolicity and Structural Stability Describing the derivative of f n(x) for different n has also been of great in- terest in the area of dynamical systems. One useful concept is that of hyper- bolicity. For that, we need the definition of the derivative on differentiable manifolds. We will employ the notation TxM for the tangent space of M at x and dxf : TxM ! Tf(x)M to be the usual linear transform between the tangent spaces, that generalize the derivative. We refer to [1] or [12] for the full definition. If f is r times continuously differentiable we will say that f is C r. The following definitions for hyperbolic sets and uniform ditto are important [1], [11]. Definition 2.1. Let f be a C 1 diffeomorphism M ! M. A compact and 4 f-invariant set Λ is called hyperbolic if for all x 2 Λ, the tangent space TxM s u s u splits as TxM = E (x)⊕E (x) where E (x) and E (x) are dfx-invariant and such that there is λ 2 (0; 1); c > 0 such that n n kdxf vk ≤ cλ kvk for all v 2 Es(x); n ≥ 1 and −n n kdxf vk ≤ cλ kvk for all v 2 Eu(x); n ≥ 1. If M = Λ fulfills the definition above, barring the condition of compactness, we say that f is uniformly hyperbolic. We call Es(x) and Eu(x) respectively the stable and unstable spaces at x. Remark 1. In the above definition it is important that c is chosen uniformly, and not point-wise. It should also be noted that there are other definitions of hyperbolicity (see Section 4), but this is a commonly used one. Example 1. Let f : R2 ! R2 be given by f(x; y) = (ax; by) where jaj > 1; jbj < 1. It then immediately follows that at each point x, we may write 2 s u s TxR = E (x) ⊕ E (x) where the stable space E (x) is the y-axis and the unstable space Eu(x) is the x-axis. Thus, f is uniformly hyperbolic. Definition 2.2. We say that a dynamical system (f; M) is topologically con- jugated to another dynamical system (g; N) if there is a homeomorphism h : M ! N, such that for all x 2 M, h ◦ f = g ◦ h. If M = N we say that f and g are equivalent. The notion of structural stability is one that is relevant for this thesis. Definition 2.3. We say that C 1 f : M ! M is structurally stable if there is an > 0 such that whenever sup kf(x) − g(x)k < , x2M sup kf 0(x) − g0(x)k < , x2M f and g are equivalent. The following theorem, first proved in [20], provides a relation between struc- tural stability and uniform hyperbolicity. Theorem 2.1. A dynamical system (f; M) where f is a C 1 diffeomorphism, is structurally stable if and only if it is uniformly hyperbolic and fulfills the strong transversality condition. 5 The exact definition of the strong transversality condition (which may be found in [23]) is quite convoluted and not crucial for this thesis. The key take-away is that in order for a dynamical system (f; M), f being a C 1 diffeomorphism, to be structurally stable it needs to be uniformly hyperbolic. 3 Lyapunov exponents While uniform hyperbolicity is and has been of great interest, it is a very restrictive trait and not every manifold admits a uniformly hyperbolic diffeo- morphism [25]. The Lyapunov exponent χf (x; v) for a function f is applicable to a wider class of dynamical systems and may be defined in the following way. Let (f; M) be a dynamical system and consider for x 2 M; v 2 TxM 1 n χf (x; v) = lim ln kdxf (v)k; (1) n!1 n if such a limit exists. If not, we may consider 1 n χf (x; v) = lim sup ln kdxf (v)k; (2) n!1 n or 1 n χf (x; v) = lim inf ln kdxf (v)k: (3) n!1 n When no ambiguity may arise, we simply write the Lyapunov exponent as χ(x; v).