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Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems

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Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems Dissertation zur Erlangung des akademischen Grades eines Doktors der Mathematik (Dr. math.) vorgelegt von Chrizaldy Neil Ma~nibo Fakult¨atf¨urMathematik Universit¨atBielefeld April 2019 Gedruckt auf alterungsbest¨andigem Papier ◦◦ISO 9706 1. Berichterstatter: Prof. Dr. Michael Baake Universit¨atBielefeld, Germany 2. Berichterstatter: A/Prof. Dr. Michael Coons University of Newcastle, Australia 3. Berichterstatter: Prof. Dr. Uwe Grimm The Open University, Milton Keynes, UK Datum der m¨undlichen Pr¨ufung:04 Juni 2019 i Contents Acknowledgementsv Introduction viii 1. Prerequisites 1 d 1.1. Point sets in R .....................................1 1.2. Symbolic dynamics and inflation rules........................1 1.2.1. Substitutions..................................1 1.2.2. Perron{Frobenius theory............................3 1.2.3. The symbolic hull...............................4 1.2.4. Inflation systems and the geometric hull...................5 1.3. Harmonic analysis and diffraction...........................7 1.3.1. Fourier transformation of functions......................7 1.3.2. Measures....................................7 1.3.3. Decomposition of positive measures.....................9 1.3.4. Autocorrelation and diffraction measure...................9 1.4. Lyapunov exponents.................................. 11 1.4.1. Lyapunov exponents for sequences of matrices................ 11 1.4.2. Matrix cocycles................................. 12 1.4.3. Ergodic theorems................................ 13 1.5. Polynomials and Mahler measures.......................... 15 1.6. Almost periodic functions and discrepancy analysis................. 16 2. Renormalisation for Pair Correlations and Absence of Absolutely Continuous Diffraction 18 2.1. Fourier matrix and inflation displacement algebra.................. 18 2.2. Pair correlation functions............................... 21 2.3. Pair correlation measures and diffraction....................... 23 2.4. Renormalisation of the Radon{Nikodym density.................. 27 2.5. Absence of absolutely continuous diffraction..................... 28 2.6. Periodic representations of quasiperiodic functions................. 30 2.7. Uniform upper bounds for χB ............................. 32 3. Constant-Length Case 36 3.1. Integer inflation multiplier: arguments in common................. 36 3.2. Binary constant-length case.............................. 38 3.2.1. Positivity of Lyapunov exponents....................... 38 ii 3.2.2. From polynomials to substitutions...................... 40 3.3. Abelian bijective case................................. 42 3.4. Mixed substitutions.................................. 49 3.5. Examples with absolutely continuous spectrum................... 52 3.5.1. Rudin{Shapiro................................. 52 3.5.2. A nine-letter example............................. 53 3.5.3. Globally-mixed examples........................... 54 4. Non-Constant-Length Case 56 4.1. Existence of exponents for irreducible Pisot substitutions............. 56 4.2. Non-Pisot examples.................................. 59 4.2.1. A family of non-Pisot substitutions...................... 59 4.2.2. Example with a Salem multiplier....................... 64 4.3. Noble means family.................................. 64 4.4. Strong irreducibility for Fibonacci Fourier matrices................. 66 5. Higher-Dimensional Examples 69 5.1. Formulation in higher dimensions........................... 69 d 5.1.1. Inflation tilings in R ............................. 69 5.1.2. Displacement and Fourier matrix....................... 70 5.1.3. Renormalisation relations........................... 70 5.1.4. Lyapunov exponents and absolutely continuous diffraction......... 72 d 5.2. Substitutions in Z ................................... 73 5.2.1. Binary block substitutions........................... 74 5.2.2. Abelian bijective block inflations....................... 76 5.2.3. Examples with absolutely continuous spectrum............... 76 5.3. Non-Pisot example: The Godr`eche{Lan¸con{Billardtiling.............. 80 6. Summary and Outlook 87 A. Dynamical Spectrum 89 B. Furstenberg's Representation 91 B.1. Random cocycles.................................... 91 B.2. Strong irreducibility.................................. 92 B.3. Approximation of stationary measures........................ 92 Bibliography 94 List of Symbols 102 List of Figures 105 List of Tables 106 Index 107 iii Abgrenzung des eigenen Beitrags gem¨aß §10(2) der Promotionsordnung (1) Most of the material in Chapter2, as well as Sections 3.3, 5.1, and 5.2.2, are part of the author's joint work with Michael Baake and Franz G¨ahler[BG¨aM18], which has been submitted to Communications in Mathematical Physics. The main results in Section 5.3 are also included in the paper, but the complete derivation appears first in this work. (2) The contents of Section 3.2 and Section 3.5.1 are included in the author's work [Man17a], which was published in Journal of Mathematical Physics in 2017. (3) The entire discussion of the non-Pisot family of inflations in Section 4.2.1 is included in the author's joint work with Michael Baake and Uwe Grimm [BGrM18] that was published in Letters in Mathematical Physics in 2018. (4) Section 3.2.2 was a part of a joint work with Michael Baake and Michael Coons [BCM17], which is to appear in the Proceedings of the Jonathan M. Borwein Commemorative Con- ference, and is currently in press. iv Acknowledgements I am eternally indebted to my supervisor and adviser, Michael Baake, for believing in me and giving me a chance to work in his group. His passion for his craft is contagious and without his suggestions, support, and patience this would not have come to reality. I also thank my other adviser, Franz G¨ahler,for sharing his expertise on a lot of things and for always being there to guide me when I feel lost. I would like to thank the current and past members of our group (in alphabetical order): Frederic Alberti, Dirk Frettl¨oh,Philipp Gohlke, Christian Huck, Tobias Jakobi, Dan Rust, Timo Spindeler, Venta Terauds, and Peter Zeiner, for fruitful discussions and interesting conversations over lunch, without which my stay in Bielefeld would have been boring. I also like to thank Herr Stephan Merkes and Frau Britta Heitbreder for enthusiastically answering my queries about administrative matters. I extend my gratitude to the German Research Foundation (DFG) for funding my studies through CRC 701 and CRC 1283. I am also grateful to Michael Coons and Uwe Grimm, for hosting me as a visiting researcher at the University of Newcastle in NSW, Australia, and at the Open University in Milton Keynes, UK, respectively. This apart from the collaboration and the enlightening chats in conferences and during their visits to Bielefeld. Credits are also due to Paolo Bugarin and Eden Miro, who paved the way for me to be able to do my PhD in Bielefeld, which is by far the most life-changing decision I made. My general view of the academic life was influenced by people who I met in conferences, or those who visited Bielefeld. In particular, I heavily benefited from discussions with David Damanik, Natalie Frank, Jeff Lagarias, and Nicolae Strungaru. I would like to specially mention Scott Balchin, who helped me with the numerics regarding Froyland's algorithm and coded a program that computes the Fourier matrix of constant-length substitutions. To Dan and Mima, no words can ever amount to all the good memories we shared and the tough days we endured together (but maybe a few pints of Weizen will). My stay in Bielefeld would not have been complete without my fellow Filipino students, with whom I have shared the gloominess of winter. To Cleo, Jay, Joseph, and Sheila: sa wakas. To the Filipino community in Bielefeld and G¨utersloh,thank you for being my home away from home, for always being thoughtful, and for always welcoming me to your homes like one of your own. I also owe a lot to my colleagues and friends in the university: the pubquiz team, Alex, Li, Marie, Omar, and Eduard. Same goes for my playmates and trainers, for volleyball games which proved to be my sanity for the last three years. To the people I met and visited while I was travelling, here's to the places waiting for us for our next escapades. Special thanks to Charles, Charlotte, Cheska, Emilie, Erika, and Quinn, v for hosting me during my trips. It goes without saying that I share this to my old friends, who knew my plight from the very beginning: Terents, Block V, Gabay, OBB, thanks for rooting for me from miles away. To my previous colleagues in the Ateneo Math department, thank you for all the support. To my previous mentors and teachers, thank you for sharing your wisdom and for the en- couragement. To my huge and wonderful family, thank you for being my source of joy and for always being there to push me to carry on. To Nikko and Pau: Doktor na si kuya. Lastly, to Mama, Papa, and Lola: I would not be where I am today without you. This is all yours. vi Para kina Cristina at Nelson, na mananatiling mas makabuluhan sa lahat ng kinaya, kinakaya, at makakaya kong isulat vii Introduction X-ray Diffraction and Aperiodic Order The landmark discovery of X-ray diffraction by von Laue and his collaborators in 1912 [FKvL12] revolutionised the state of non-contact characterisation methods of materials. In that work, it was confirmed that X-rays have wavelengths which are compatible with atomic spacing in solids which allows one to consider solids
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