Spectral gap and asymptotics for a family of cocycles of Perron-Frobenius operators
by
Joseph Anthony Horan MSc, University of Victoria, 2015 BMath, University of Waterloo, 2013
A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
in the Department of Mathematics and Statistics
c
Joseph Anthony Horan, 2020 University of Victoria
All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author.
We acknowledge with respect the Lekwungen peoples on whose traditional territory the university stands, and the Songhees, Esquimalt, and W
¯S ´ANE ´C peoples whose historical relationships with the land continue to this day.
Spectral gap and asymptotics for a family of cocycles of Perron-Frobenius operators
by
Joseph Anthony Horan MSc, University of Victoria, 2015 BMath, University of Waterloo, 2013
Supervisory Committee
Dr. Christopher Bose, Co-Supervisor
(Department of Mathematics and Statistics)
Dr. Anthony Quas, Co-Supervisor
(Department of Mathematics and Statistics)
Dr. Sue Whitesides, Outside Member (Department of Computer Science)
ABSTRACT
At its core, a dynamical system is a set of things and rules for how they change. In the study of dynamical systems, we often ask questions about long-term or average phe-nomena: whether or not there is an equilibrium for the system, and if so, how quickly the system approaches that equilibrium. These questions are more challenging in the non-autonomous (or random) setting, where the rules change over time. The main goal of this dissertation is to develop new tools with which to study random dynamical systems, and demonstrate their application in a non-trivial context. We prove a new Perron-Frobenius theorem for cocycles of bounded linear operators which preserve and sometimes contract a cone in a Banach space; this new theorem provides an explicit up-per bound for the second-largest Lyapunov exponent of the cocycle, which determines how quickly the system approaches its equilibrium-like state. Using this theorem and other tools (including a new Lasota-Yorke-type inequality for Perron-Frobenius opera-tors for use with a family of maps), we show that a class of cocycles of piecewise linear maps has a Lyapunov spectral gap (hence answering the equilibrium question in the affirmative), and we moreover have an explicit lower bound on the spectral gap. We also prove asymptotics for a family of cocycles arising from a perturbation of a fixed map with two invariant densities; we obtain a linear upper bound for the second-largest Lyapunov exponent, and the bound is sharp, in the sense that there are members of this family of perturbations where the second-largest Lyapunov exponent is linear in the perturbation parameter. The sharpness example is studied through an in-depth determinant-free linear algebra computation for Markov operators.
Table of Contents
Supervisory Committee ii Abstract iii Table of Contents iv List of Figures vi Acknowledgements viii Dedication xiv Chapter 1 Introduction 1Chapter 2 Cocycle Perron-Frobenius Theorem 12
2.1 Cones . . . 13
2.2 Measurability and Topological Considerations . . . 30
2.3 Cocycles, Lyapunov Exponents, and the Grassmannian . . . 33
2.4 Statement of the Main Theorem . . . 36
2.5 Proof of Theorem 2.4.3 and Corollary 2.4.5 . . . 39
2.6 Easy Applications of Theorem 2.4.3 . . . 52
Chapter 3 Balanced Lasota-Yorke-type Inequality 59 3.1 Bounded Variation and Setting . . . 60
3.2 Statement and Proof of the Inequality . . . 77
Chapter 4 Application to Cocycles of Perron-Frobenius Operators 91 4.1 Cyclic Decomposition . . . 94
4.2 Uniform Lasota-Yorke Inequality . . . 97
4.3 Covering Properties . . . 104
4.5 Contraction of the Cone - Spectral Gap for L(n)ω . . . 115
4.6 Perturbation Asymptotics . . . 126
Chapter 5 Markov Paired Tent Maps 136 5.1 Markov Maps and Partitions . . . 136
5.2 Spectral Properties of An . . . 141
5.3 Spectral Properties of a Factor System . . . 149
5.4 Mixing Rates and Times . . . 153
5.5 Simultaneous Spectrum via Algebraic Geometry . . . 155
5.6 Two-Parameter Markov Paired Tent Maps . . . 160
Chapter 6 Conclusion 164 Appendix A Assorted Lemmas, Proofs, and Computations 166 A.1 Miscellaneous Ergodic Theory . . . 166
A.2 Miscellaneous Tools . . . 170
A.3 Miscellaneous Examples . . . 175
List of Figures
Figure 1.1 A Markov chain with four states and its associated transition matrix. . . 2 Figure 1.2 The stirring map T that moves chocolate chips around in the
banana bread batter [−1, 1]. . . 3 Figure 1.3 The Perron-Frobenius operator L sends densities f to new
den-sities Lf (schematic only). . . 4 Figure 1.4 Schematic of “leaking” behaviour, where 1(ω) and 2(ω) are
gen-erally small. . . 9 Figure 2.1 The cone Cy,α for y = (1, 2) and α = 2 in R2, depicted with the
perpendicular plane at cy with c = 1/2. . . 18 Figure 2.2 Schematics of cocycles along an orbit of x (both non-invertible
and invertible). . . 33 Figure 3.1 FH(f ), for f (x) = 36x3+ 4x2− 3x − 0.1 restricted to [−1, 1] and
H = [−0.4, 0.3]. . . 72 Figure 3.2 The setup in Example 3.2.6. . . 85 Figure 3.3 Computing L(f ) using Lemma 3.2.4. . . 86 Figure 3.4 A map T : [0, 1] → [0, 1] with four hanging points, namely:
(0.4, −), (0.4, +), (0.6, +), and (1, −). . . 87 Figure 4.1 The paired tent map, with parameters 1 = 0.3 and 2 = 0.7. . . 92
Figure 4.2 The second iterate of the coupled tent map, Sω, with parameters
1(ω) = 0.1, 2(ω) = 0.2, 1(σ(ω)) = 0.1, and 2(σ(ω)) = 0.2. . . 98
Figure 4.3 The second iterate of the coupled tent map, Sω, with parameters
1(ω) = 0.7, 2(ω) = 0.3, 1(σ(ω)) = 0.2, and 2(σ(ω)) = 0.6. . . 99
Figure 4.4 The function f and its image under the Perron-Frobenius op-erator Pω. Observe that both functions are supported only on
[0, 1]. . . 105 Figure 4.5 The map T in Remark 4.3.6. . . 109
Figure 4.6 A picture of (Lδ1,δ2 − L1,2)1[−1,−1/2] for 0 < δ1 < 1. The first
jump is of size 2(1+δ1
1), and the second jump is of size
1 2(1+1), so
the variation is the sum of the two jump sizes. . . 126 Figure 5.1 The paired tent map Tκ,κ, with parameter κ = 0.3. . . 137
Figure 5.2 Markov partitions for Tκn, with n = 1, 4. . . 140
Figure 5.3 General form of the (2n + 4)-by-(2n + 4) adjacency matrix An. . 141
Figure 5.4 The (2n + 4)-by-(2n + 4) matrix Jn. . . 142
Figure 5.5 A zoomed-in look at the Markov partition for Tn in [−1, 0]. . . . 143
Figure 5.6 Pictures of the roots of fn and gn for different values of n; roots
of fnare marked with crosses, roots of gnare marked with circles,
and the origin is marked with an asterisk (where Anhas a double
eigenvalue). The circle of radius 2 is a dashed line, the unit circle is a solid line, and the circles with radius 1 ± n−1 are dotted lines.147 Figure 5.7 The map ˜Tκ, for κ = 0.3. . . 149
Figure 5.8 Markov partitions for ˜Tn, for n = 1, 4. . . 151
Figure 5.9 General form of the (n + 3)-by-(n + 3) adjacency matrix Bn. . . 151
Figure 5.10 The (n + 3)-by-(n + 2) matrix ι, representing the inclusion E+ →
Cn+3. . . 151 Figure 5.11 The (n + 2)-by-(n + 2) matrix Cn, representing the action of An
on E+. . . 152 Figure 5.12 The steeper function (in blue) is dm(κ) and the flatter function
(in red) is cn(κ), for m = 2 and n = 3. The x-coordinate of
the intersection is the κ value corresponding to (n, m) and the y-coordinate is the ζ value. . . 161
ACKNOWLEDGEMENTS
Those who have spent significant time with me over the course of my doctorate will not be surprised to see that my list of acknowledgements is a metaphorical mile long (for the purpose of using metric units, it’s actually about 1.15 metres, give or take). For my Master’s, I wrote no names; all of the people I am about to name deserve to be recognized for being wonderful people and making my life better in one way or another during my Ph.D. (Note that folks will only be named once, even if they played multiple roles in my life.) If I forgot anyone, that’s on me and not on you.
First and foremost, I would like to thank my supervisors, Christopher Bose and Anthony Quas. It has been an interesting six years and eight months! Thank you for being fabulous academic mentors. Your willingness to meet with me often, talk about whatever, and be candid with me has been a great help. It is striking how much more I am like Chris than Anthony, isn’t it? But I learned so much from both of you regardless. I hope you two appreciated the time we spent together bashing our heads on the chalkboard in the lounge and bothering everyone on the fifth floor with our Skype conversations; I did some good work thinking on the fly, I think. I admit that I will appreciate not having to look your comments on my work for a while, though. I should also express my appreciation for your financial support, through NSERC (in addition to my own scholarship).
Next, I would like to thank two people who have very little to do with my re-search and much more to do with my teaching and mental health, Jane Butterfield and Christopher Eagle. I will not forget how, back in 2014, you e-mailed me, Jane, asking about how on my webpage, I said I wanted to help folks learn how to mark better (read: without being miserable). I’m not sure I figured that one out, but anyway. Your open door and willingness to listen meant that it was probably helpful for you that I moved upstairs when I started my Ph.D. The vast amount that I learned about teaching from you is no less important than how much I learned about research over the years. In addition, thank you for making the Assistance Centre one of the places in which I most enjoyed being; it really felt like home. Chris, thank you (as well) for teaching me about teaching, but perhaps I appreciated even more your interest in research and your enthusiasm for sharing fun problems and interesting theorems. I know I jump straight to guessing that whatever you’re talking about has to do with
the Riemann or Continuum Hypotheses, but still! I am also glad that I was able to show you how Eisenstein’s Criterion actually has a use in the wild; you’ll keep that one with you forever.
I would like to give a blanket statement of thanks to the entire office staff in the Math and Stats department: Carol Anne Sargent, Amy Almeida, Patti Arts, and Kristina McKinnon (as well as the others who filled some of those spots throughout the years). I haven’t always been the model student in terms of finishing things much before deadlines (sorry, Amy), but you folks have always been great about keeping the department moving and setting me straight on whatever it is I’m confused about. You’re all amazing. The same goes for Kelly Choo, our systems administrator: I could always count on a good answer from you about all of my tech concerns.
There is a long list of the other faculty members in the department with whom I worked and had great conversations; in alphabetical order by last name, thanks to Trefor Bazett, Peter Dukes, Marcelo Laca, Mary Lespersance, Gary MacGillivray, Kieka Mynhardt, Svetlana Oshkai, and Ian Putnam for being wonderful individuals. Also thanks to Chedo Barone, who was not faculty but was one of my favourite people in the department: your smile always brightened my day.
At this point, I would like to turn my attention to some of my friends in Victoria. I found it was hard to make friends here, but these people gave me time and for that, I value them immensely. Julie Fortin, you are sharp, on-the-ball, and quick to smile; you made an amazing Director of Services and my time spent with you was always fun. Thank you, also, for actually reading this document! Janet Sit, I always appreciated your wit and your love of music (and also of science!). Thank you for running Trivia with me and hanging out afterwards; thank you for having adorable stuffed animals and an insatiable desire to learn. You are so courageous; never forget that. Alyssa Halpin, thank you for the hugs, primarily, but also thank you for the conversations and the perspective and your willingness to listen. It has always been nice to hang out with you (and Garrett Culos! I miss him). Alyssa Allen, I appreciated your smile and quirkiness and the fun times we had hanging out doing work or playing board games (when we could make it happen). You certainly helped me learn not to worry about texting. Elissa Whittington, thank you for the lunch hangouts and casual conversation; I’m glad that we could always find something to talk about over so many years.
I spent a lot of time volunteering for the UVic Graduate Students’ Society (the GSS, in the common lingo) over my doctorate program; probably way more than reasonable, but here we are. Thank you to all of the Executive Board members who did such a great job keeping the GSS headed in the right direction (there are a lot of you). A special thank you goes to the lovely staff! Stacy Chappel, you have always been a fantastic resource and a good friend; you care so much. Karen Potts, I hope you find someone on Grad Council who cares about minutes and governance as much as I did. Neil Barney, you are such a fabulous person and we saw eye-to-eye on a lot of things; my Events Committee experience would not have been nearly as great without you. Jo¨elle Alice Michaud-Ouellet and Mindy Jiang, you have done such a great job with the Health and Dental Plan administration and you always had a smile for me in the office. Shout-outs to Rachel Lallouz; I always enjoyed seeing you in the office. Thank you to all of my Stipend Reviewers over the years; I especially liked this past year’s edition, featuring my favourite cowboy-hat-wearing Nicholas Planidin. Thank you also to my button-making buddy, Brooklynn Trimble; thank you for taking me up on that opportunity and hanging out with me at other times (and giving the GSS reason to replace the circle cutter; glad you’re okay). Here, I would also like to thank all of the people who came out regularly to my Trivia events: Tiffany Chan, Rose Morris, and Melanie Oberg from English, Kate Fairley and the Econ folks (with their silly team names), and many others.
I am very thankful that the Math and Stats department had a graduate student group: SIGMAS made it worth trying to organize events. Thank you to all of the people who enjoyed reading my Tea-Mail and eating my baked goods at Tuesday Tea! That was always a highlight for me. Thank you to my officemates who endured my loud keyboard and still talked to me every once in a while; special thanks go to my favourite big sister that I never had, Joanna Niezen. I appreciated all of our conversations and teasing and jokes and shared experiences; I’ll take good care of Claire Pollenegger (our fake office plant, for those who didn’t stop by). Thank you to the various members of the SIGMAS Executive who tolerated my insistence on minutes and whatnot, es-pecially Laura Teshima; you are a wonderful person. Thank you to all of the people who made Chris Bruce’s learning seminars fun: among others, Anna Duwenig, Mark Piraino, Emily Korfanty, Dan Hudson, Anthony Cecil, and Dina Buric. Thank you also to the folks who made the CMS Math Camps happen, at various times, including Amanda Malloch and Brittany Halverson-Duncan; we’ll have to do pizza again when the world stops ending. I’ll throw out some other names here for folks I appreciated,
no matter what you role you had: Flora Bowditch, Mackenzie Wheeler, Chloe Lamp-man, Felicia Halliday, MacKenzie Carr, Kevin Hsu, Chi Kou, Sam Churchill, Jane Wodlinger, Michelle Edwards, Josh Manzer, and Kseniya Garaschuk.
I spent a large amount of time at workshops hosted by the Division of Learning and Teaching Support and Innovation (LTSI), which houses the former Learning and Teaching Centre. I value what I learned there, but at the same time I hope I was helpful to others at those same workshops (especially when I attended the opening session of the Fall TA Conference for the fifth, sixth, or seventh straight year). I especially want to thank Gerry Gourlay; your endless positivity and cheerfulness is infectious. I would not be as comfortable with the whole idea of intended learning outcomes without our conversations.
My physical health would like to thank the members of the Physics and Astronomy (and friends) intramural ball hockey team with whom I played for a few years; we weren’t that great, but we sure tried hard and had fun. I particularly remember Sandra Frey, Nick Fantin, Ashley Bramwell, Jared Keown, Collin Kielty, our two Clares (Higgs and Trotter), Jemma Green, Emma Loy, Zack Draper, Maan Hani, Ben Gerard, and Tony Kwan. I am so thankful that you welcomed me so readily.
I would like to make a point of thanking Jennifer Wong and Mira Cvitanovic for doing such a great job putting Convocation together; it was such a delight to volunteer as a robing assistant. I look forward to reprising that role while wearing my own robes, at some point!
I had the absolute pleasure of volunteering with UVic Orientation for many years, including this last year as the Graduate Tour Leader Cohort Lead. I have often said that the best opportunities for volunteering are when the people with whom you are working are fantastic; here is no exception. I want to thank, in particular, Kate Hollefreund, Jasmine Peachey, Nora Loyst, and Suriani Dzulkifli for being positive, enthusiastic, and incredibly well-organized facilitators of the Orientation program; I enjoyed working with you immensely, especially because you welcomed my willingness to help throughout the entire day of Graduate Student Orientation year after year. I also want to thank Russ Wong for keeping in touch and giving me an outlet for shared experiences; there aren’t many folks who know what Orientation is like at both UVic and the University of Waterloo, but here are two of us!
I had the pleasure of being involved in Three Minute Thesis and the Faculty of Graduate Studies Council. In particular, thank you to Carolyn Swayze, Bernadette Perry, and Karolina Papera Valente for all of your help in 3MT, and thank you to David Capson for doing such a great job as Dean for all of these years. I was also involved in the inaugural President’s Fellowship in Research-Enriched Teaching; I am glad to have spent so much time with the other recipients, including Stephanie Field, Mary Anne Vallianatos, and Carla Osborne.
Lastly among my Victoria friends, I would like to acknowledge my Learning And Teaching in Higher Education (LATHE) 2018-2020 cohort: Janice Niemann, Natalie Boldt, Elizabeth Williams, Tasha Jarisz, Mitch Haslehurst, Pierre Iachetti, Mohamed Seifeldin, Jeremy Wintringer, and H´ector V´azquez (as well as Les Sylven, who was only with us for a term). Our classes felt like home to me. I will forever be indebted to you for your kindness and your wisdom; thank you for being my friends. I will always be there for you. I promise.
Moving out west in 2013, I left behind an entire undergraduate degree’s worth of support network, but I still have many friends out in Ontario whom I still hold very dear (as evidenced by the cookies I have mailed across the country). Thank you to Michelle Cannon, Sandra Regier, Kimberley McClatchie, Heather (n´ee Isenegger) and Michael Overmeyer, Christopher Snow, Sophie Twardus, Robin Lawrence, Paul Hendry, and Emma McCutcheon for being such wonderful people; there’s a reason I always try to visit when I pass through. Special thanks to Paul and Emma (and their dog Arlo!), who made life in Victoria for me that much better while they were living there too.
I have three friends out east about whom I would like to be more specific. Thank you to Carolyn Kimball, for continuing to talk with me even after going to Switzer-land and back; I believe in your ability to succeed in whatever you are doing. You are such a caring person. Thank you to Katie Schreiner; you have a fantastic taste in church services and music, and I have appreciated all of the time we have spent together. Maybe we’ll write more songs together, sometime? If you don’t get to be an astronaut, then you’ll still get to canoe and play your ukelele (okay, and probably do some cool river dynamics); I hope that’s something. And perhaps above all, thank you to Melissa Snow (n´ee Pettau): you have always been there for me, even if only as a string of characters on a screen. I will always be grateful for our conversations and your friendship.
Finally, thank you to my extended family. I moved to Victoria partially because I had family in town; I have loved all of the joint birthday celebrations and playing baseball and eating amazing dinners. I could not have asked for better family. Thank you also to my brother, Jeffrey: though we don’t talk that much, we have a much better relationship than we did back when we were in high school. We should chat more; maybe one day I’ll come to visit you.
DEDICATION
My dissertation is dedicated to my parents, Dan and Lina. Whether I have noticed or not, they have always been supportive of me and have never once pushed me to do the things I have ended up doing. They let me grow and figure things out on my own and helped me up whenever I tripped and fell. I probably would not have ended up here without them; in particular, I am glad that I realized that I still know their landline number by heart, because while I just listed a whole bunch of people with whom I have enjoyed interacting, my parents are the only people I know who are unconditionally willing to take phone calls about anything. Incidentally, they have also been fantastic landlords. I really did luck out, no matter what they say. I hope they are proud that I got this far.
Chances are, I have not said the following enough in my life, so I will write it in the most conspicuous space in the entire document:
Chapter 1
Introduction
Dynamical systems are, whether people are aware of it or not, everywhere around us. At a very high level, a dynamical system is some set of things (maybe states of an object, or gas particles in box; maybe the Moon revolving around the Earth, or the tides on the beach) together with rules for how these things change. It is natural to ask questions about these systems: what will usually happen, and how long does it take? Does the system tend to an equilibrium state, like a chemical reaction or a simple spring? What happens if, instead of a fixed set of rules, the rules change over time; can we still expect the system to tend to some sort of equilibrium at some rate? For a natural system, we find mathematical models to describe the system and apply various methods to answer questions about the system’s properties. For certain models, we may even need to first develop new tools in order to answer more than the simplest questions about the model. The contribution of this dissertation is exactly that: new tools developed to study an interesting model of dynamical systems.
Markov Chains and Deterministic Dynamical Systems
Consider, for the time being, a column stochastic d-by-d matrix P . The matrix P represents a finite-dimensional Markov chain, a stochastic model where states transition to one another with some probability at discrete time steps according to the entries in the matrix. Thus, if at time 0 the probabilities of being in each of the d states are given by the vector x, then the probabilities of being in each of the d states at time 1 are given by P x (P acting on x); see Figure 1.1. The matrix P is called the probability transition matrix for the Markov chain. We already know how to study finite-dimensional Markov chains: the asymptotic properties of the Markov chain, such as what the stationary distribution is, are determined by the spectral theory of the matrix P . Tools arising from linear algebra, potentially including numerical computation techniques, then allow us to compute these desired quantities: the stationary distribution is the eigenvector corresponding to the eigenvalue 1, for example.
1 2 3 4 0.3 0.2 0.5 0.6 0.4 0.4 0.4 0.4 P = 0.5 0.2 0.1 0.1 0.3 0.6 0.1 0.1 0.1 0.1 0.4 0.4 0.1 0.1 0.4 0.4
Figure 1.1: A Markov chain with four states and its associated transition matrix.
stationary distribution to which all initial distributions converge), we wish to find the modulus of the second-largest eigenvalue(s), which tells us the rate at which the Markov chain converges to the stationary distribution. The mixing time for the chain is then at most proportional to the reciprocal of the logarithm of the modulus of the second-largest eigenvalue.1 To prove that a Markov chain is mixing, if the transition matrix P
is primitive, meaning that there is a power of P that has all positive entries, then we can use the classical Perron-Frobenius theorem [15, 42] to show that the chain has a unique stationary distribution and a second-largest eigenvalue of modulus strictly less than 1.
How does the Markov chain model fit into our dynamical systems perspective? Instead of seeing the chain as a stochastic process, we can view the elements of the system as the probability vectors, and the elements change via multiplication by the probability transition matrix, so that the rule for change is simply multiplication by P . We could even look at all vectors in our finite dimensional space instead of just those with non-negative entries summing to 1; then we really are looking at the action of a (bounded) linear operator on a vector space. As mentioned above, we certainly have tools for analysis of this situation!
Returning to the general dynamical systems setting, if we have a map T on some state space X, some questions we would like to answer are “what happens to most of the orbits of T over a long time?” and “do regions of X mix together over time, and at what rate?” These questions are less about looking at individual orbits of points under T and more about looking at what happens on average. Specifically, we can learn much about the dynamical system (X, T ) by studying how probability densities on X change over time under the action of T . If our state space X is finite, then this feels very familiar; it is the Markov chain situation again! But if X is more general, like an interval or a riverbed or an ocean, then the densities (probability or heat or
1For the proof of this fact and for more on Markov chains, see the book by Levin, Peres, and
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1
Figure 1.2: The stirring map T that moves chocolate chips around in the banana bread batter [−1, 1].
similar) on X no longer form a subset of a finite-dimensional vector space; they are now infinite-dimensional, and we must find new tools to perform the analysis that we desire.
As both a mostly-concrete example and a rough analogy (and to introduce the key example that we will explore in much greater detail later), consider the space X = [−1, 1] equipped with normalized Lebesgue measure λ, and let f ∈ L1(λ) be a
probability density (that is, kf k1 = 1 and f ≥ 0). We can imagine that the space [−1, 1] is a bowl of banana bread batter into which one has placed chocolate chips, and f is the density of chocolate chips. Then, let T be a map like the one pictured in Figure 1.2. Applying the map T stirs the space up like you would with a spoon or a mixer, moving the chocolate chips around; there is then a new density, call it Lf , that describes the new locations of the chocolate chips; see the schematic diagram in Figure 1.3. Some parts of the batter may have more chocolate chips than before, and some fewer, but the total amount of chocolate chips has not changed. If we continue to apply the map T to get, after n steps, Tn, then we should get something like Lnf ,
a new density that describes where the chocolate chips are after n steps.
It turns out that the operator L can be defined on all of L1(λ), and is bounded and
linear; we call L the Perron-Frobenius operator associated to T . To be rigorous, Lf is the Radon-Nikodym derivative of the measure A 7→ λ(f 1T−1(A)), which exists in this
case because T is a non-singular map (if A has zero measure, then T−1(A) also has zero measure); see, for example, Chapter 4 of [7]. Now it really does look like we are doing
f L Lf
Figure 1.3: The Perron-Frobenius operator L sends densities f to new densities Lf (schematic only).
something similar to the Markov chain case; we have a Banach space and a bounded linear operator acting on the space in a physically meaningful way. The operator L is the infinite-dimensional analogue of the transition matrix P for the Markov chain. If we want to find the analogue of a stationary distribution (called an invariant density in this setting), we find an eigenvector for L with eigenvalue 1; if all initial densities on [−1, 1] converge to an invariant density over time, then we have a good idea of where most of the points in [−1, 1] end up in the long run: no matter where they started, points will be distributed over [−1, 1] according to the invariant density. Moreover, if there is a gap in modulus between an eigenvalue of 1 and the rest of the spectrum, this gap describes how quickly this convergence occurs, in the same way as described above for Markov chains. Note that if the map T from Figure 1.2 had tents that did not cross the x-axis, then T would leave the regions [−1, 0] and [0, 1] invariant; it would be as though you separated the cookie batter into two pieces and only mixed each piece with itself. It is easy to imagine that in this case, T would have two invariant densities: the characteristic functions on [−1, 0] and [0, 1], respectively.
Unfortunately, some technical details get in the way of the analysis; the spectrum of L on L1(λ) is not particularly well-behaved in general,2 so we find an invariant subspace of L1(λ) on which the spectrum of L is much nicer, and then the analysis
follows basically as in the Markov chain case, using more powerful and sophisticated tools. The desired subspace is that of the (equivalence classes of) bounded variation functions, which we will denote by BV (λ). Lasota and Yorke [29] showed, in 1973, that for a class of piecewise-expanding maps, one can find invariant densities by looking at bounded variation functions and using their namesake inequality; in 1984, Keller [26] discussed the link between the spectral theory of Perron-Frobenius operators acting on bounded variation functions and the rate of convergence of the underlying dynamical systems to an equilibrium. Other authors have used the compact embedding of BV inside of L1 to perform similar types of spectral analysis (see Hennion [20] or Keller
and Liverani [27]).
Analogously to the classical Perron-Frobenius theorem in the Markov chain setting, there are similar general theorems, often called Krein-Rutman theorems after the
gen-2As shown by Ding, Du, and Li [9], Perron-Frobenius operators can have L1-spectrum equal to the
eralization of the Perron-Frobenius theorem for compact positive operators by Krein and Rutman [28], that assist in analysis of these dynamical systems. The key aspect in all of these theorems is the notion of positivity. Roughly, the positive direction is indicated by a cone of vectors, and operators which preserve and contract the cone have very nice structure: there is a vector in the cone playing the role of the direction of largest growth, and a complementary subspace of non-positive vectors which grow more slowly than anything in the positive direction. This insight was notably abstracted by Garrett Birkhoff in the 1950s [5], and the study of abstract cones has played a role in the study of general topological vector spaces (for example, [41, 49]). In 1995, Liverani [33, 34] applied Birkhoff’s technique to the study of piecewise-expanding C2 dynamical
systems, obtaining invariant densities and explicit decay of correlations for powers of a single map.
Random Dynamical Systems
We can then take this idea one more step forward. Typically, when mixing banana bread batter, you would not always mix it the same way. This idea corresponds to applying not just a single map T to [−1, 1] at each step, but to apply a map chosen from a family of maps Tω, parameterized by some set Ω, with a different map chosen
at each time step. When we look at the composition of a number of these maps, it gives us a notion of random dynamical system; the adjective is appropriate, at least by analogy with the batter mixing, because you do not know exactly how you will have mixed the batter after n steps.
For a more rigorous formulation, we turn to abstract measurable dynamical systems theory. Let µ be a probability measure on the set Ω, and suppose that σ : Ω → Ω is a measurable map such that (µ, σ) is an ergodic map-and-measure pair; for short, we say that (Ω, µ, σ) is an ergodic probability-preserving transformation (taking the underlying σ-algebra on Ω for granted). Suppose that there is a measurable assignment of some map Tω to each ω in Ω. Then we can define the n-th step map T
(n) ω to be
Tσn−1(ω) ◦ · · · ◦ Tω, where we compose the maps Tσi(ω) along the orbit of ω under σ.
Since (Ω, µ, σ) is ergodic, the map Tω(n) is random, in the sense that on average in the
long run, each of the individual maps are chosen roughly according to the probability distribution µ (which may not be true for short time scales). We call the map Tω(n)
a cocycle of maps. We will be assuming that σ is also invertible throughout, so that (Ω, µ, σ) is an ergodic invertible probability-preserving transformation.
We are interested in the long-term behaviour of the random dynamical system, just like for our deterministic systems from before. Is there something like a random stationary distribution? Do densities converge at some rate to the stationary distribu-tion; if so, what is the rate? Unlike before, we observe that it makes no sense to try to consider eigenvalues of associated Perron-Frobenius operators L(n)ω for Tω(n); since the
operators L(n)ω are also compositions of the individual operators Lσi(ω) along the orbit
of ω (hence a cocycle of operators), the eigenvalues would depend on ω and n and could easily be different for each pair in all but the simplest cases. Instead, we have different quantities, called Lyapunov exponents, that play the same role as the eigenvalues, but focus more on the asymptotic and on-average behaviour of the maps Tω(n); instead of
eigenspaces, we have Oseledets spaces, named for the first person to introduce them (in [40]). The associated Perron-Frobenius operators are invariant on these subspaces, and functions (densities) in these subspaces grow at specific rates corresponding to the Lyapunov exponents.
The framework for studying the actions of these random compositions of linear op-erators is multiplicative ergodic theory, named after Oseledets’s Multiplicative Ergodic Theorem (MET) for cocycles of matrices in [40]. There is a rich literature of general-izations of the original MET, first to cocycles of compact operators on Hilbert spaces [46], then to cocycles of compact operators on Banach spaces [35], and then to quasi-compact cocycles of bounded operators on Banach spaces with varying measurability and continuity requirements [31, 50]; in every case, the base system (Ω, µ, σ) and the matrices or operators are invertible. More recently, these generalizations have been extended [16, 17, 19] to allow for no invertibility conditions on the operators while retaining the full decomposition of the Banach space into Oseledets spaces, as long as the base system (Ω, µ, σ) remains invertible (previously, for non-invertible operators one only obtains a “flag” of subspaces on which vectors grow at a rate at most one of the Lyapunov exponents, instead of exactly at that rate). An MET can be seen as the replacement for diagonalizability of a single matrix in the cocycle setting (although as many of the proofs indicate, it takes on more of a singular value decomposition flavour, as seen in [16, 44]).
All of this said, one does not need an MET to define Lyapunov exponents, just as one does not need diagonalizability to define eigenvalues, so we can see how other tools play a role and interact with the MET in a given situation. In particular, is there some sort of generalization of the classical Perron-Frobenius theorem or Krein-Rutman theorems for cocycles or matrices or operators with some notion of positivity, that would guarantee the existence of something like an invariant subspace corresponding to the largest growth rate? The answer: yes! (Though the subspaces are called equivariant rather than invariant.) The literature is fairly extensive on this subject; we give a sampling. In 1988, Ferrero and Schmitt [14] proved a cocycle Perron-Frobenius theorem for cocycles of transfer operators on symbolic dynamics spaces. In 1994, Arnold, Demetrius, and Grunwald [2] proved a cocycle Perron-Frobenius theorem for a class of positive matrix cocycles arising in evolutionary biology, based on the Birkhoff cone technique, for the purpose of obtaining a random invariant density. Hennion [21], in 1997, used positivity to establish properties of stochastic products of matrices. Evstigneev and Pirogov [13] established a cocycle Perron-Frobenius theorem for non-linear operators in finite
dimensions in 2009. In 2010, Rugh [47] developed the theory of complex cones and proves a cocycle Perron-Frobenius theorem for linear operators preserving complex cones (as opposed to real cones, in real Banach spaces). Mierczy´nski and Shen [39], in 2013, proved a Perron-Frobenius theorem for cocycles of compact operators. In 2015, Lian and Wang [32] prove a cocycle Perron-Frobenius theorem for finite-dimensional operators preserving cones of specific lower dimensions.
When we apply Multiplicative Ergodic Theorems to quasi-compact cocycles of Perron-Frobenius operators arising from random dynamical systems, the Oseledets spaces can be interpreted as indicating “coherent structures” in the system, as de-scribed in [16]. When the spaces correspond to large Lyapunov exponents of the cocycle (when compared to local expansion and dispersion), the Lyapunov exponents describe how these parts of the system are “slowly exponentially mixing”, in the sense that they are not invariant sets but they mix with the rest of the space more slowly than one would expect from local expansion. When the Oseledets space corresponding to the zero Lyapunov exponent turns out to be one-dimensional, there is a complementary equivariant family of subspaces on which the Perron-Frobenius operator cocycle, re-stricted to the spaces, has the second-largest Lyapunov exponent. Thus, if we are able to compute a gap between the largest and second-largest Lyapunov exponents, then we can show that all dissipating coherent structures mix with the space at least at some rate.
Specifically for cocycles of Perron-Frobenius operators, Buzzi [8] in 1999 extended Liverani’s application of Birkhoff’s positive operator framework to cocycles of maps, instead of a single map, to obtain decay of correlations for certain random dynamical systems where not every Perron-Frobenius operator is required to preserve a cone. In this case, the largest Lyapunov exponent is equal to 0 and corresponds to an equivariant family of densities. Then, the decay of correlations is related to the second-largest Lyapunov exponent for the cocycle of operators, and its magnitude gives the logarithm of the rate of decay of correlations. However, the constants involved in the proof of Buzzi’s result are not easily identifiable (see the remark on pg. 28).
Summary of Results
As outlined previously, we are therefore interested in finding an upper bound for the second Lyapunov exponent for the Perron-Frobenius cocycle corresponding to a random dynamical system Tω(n), as that would give us a minimal mixing rate for the system (or
decay of correlations). The approach we take is to prove a generalized Perron-Frobenius theorem for a fairly general class of cocycles of operators on a Banach space preserving a cone. In particular, no compactness is required (as in [39]); however, we do require that almost every operator preserves the cone (which is a stronger restriction than what Buzzi works with in the specific cases in [8]). The setup we use allows us to obtain a
measurable equivariant decomposition of the Banach space into an equivariant positive direction with the largest growth rate and an equivariant family of subspaces of non-positive vectors growing at the next largest growth rate. An important outcome of the theorem is a rigorous quantitative bound on the second-largest Lyapunov exponent for such cocycles that is computable without tracing constants along in the proof of the theorem itself. Another important aspect of the theorem is that the proof is completely independent of any Multiplicative Ergodic Theorem; hence, the hypotheses provide a checkable condition for quasi-compactness and thus a full Oseledets decomposition for the cocycle after applying an MET in the appropriate setting. A summarized version of the theorem follows; see Section 2.4 for more details, including required definitions and what is meant by measurable in our context. Moreover, Corollary 2.4.5 provides even simpler sufficient conditions for the existence of the quantities listed here in the hypotheses.
Theorem A. Let (Ω, B, µ, σ, X, k·k , L) be either a strongly measurable random dynam-ical system or a µ-continuous random dynamdynam-ical system, such that log+kL(1, ·)k ∈ L1(µ). Let C ⊆ X be a nice cone such that L(1, ω)C ⊆ C for all ω. Suppose that
there exists a positive measure subset GP of Ω, a positive integer kP, and a positive
real number DP such that for all ω ∈ GP, diamθ L(kP, ω)C ≤ DP. Then there exists
a σ-invariant set of full measure ˜Ω ⊆ Ω on which the following statements are true: 1. There exist measurable functions v(ω) ∈ X and η(ω, ·) ∈ X∗ such that
X = spanR{v(ω)} ⊕ ker(η(ω, ·))
is a measurable equivariant decomposition, the Lyapunov exponent for v(ω) is λ1, and all vectors in ker(η(ω, ·)) have Lyapunov exponent strictly less than λ1,
unless λ1 = −∞.
2. When λ1 > −∞, we have λ2 ≤ λ1−
µ(GP)
kP
log tanh(14DP)−1 < λ1.
We remark specifically on the quantitative bound on the second-largest Lyapunov exponent. The set GP and the quantities kP and DP are dependent on the cone and
the cocycle and can therefore be computed outside of the proof of the theorem. The theorem is therefore something of a black box for the bound and, subsequently, a minimal mixing rate or decay of correlations. Outside of cocycles of positive operators, this problem is quite difficult.
To demonstrate the use of the theorem, we apply it to the situation of a cocycle of piecewise expanding maps with a specific form; all of the maps Tω are of the same
form as the map pictured in Figure 1.2. These “paired tent maps” act on [−1, 1] and leave [−1, 0], [0, 1] mostly invariant, except for “leaking” mass of size 1(ω) ≥ 0
[−1, 0] [0, 1] 1(ω)
2(ω)
Figure 1.4: Schematic of “leaking” be-haviour, where 1(ω) and 2(ω) are
gener-ally small.
for the precise definitions and Figure 1.4 for a simple schematic diagram). Maps like these have been considered by Gonz´alez Tokman, Hunt, and Wright in [18]; in that work, the authors fix a single perturbation of a map that leaves two sets invariant and investigate properties of the invariant density and the eigenvector for the second-largest eigenvalue for the perturbed map, in terms of the two invariant densities for the unperturbed map. They find that the eigenvector corresponding to the second-largest eigenvalue is asymptotically a scalar multiple of the difference of the two invariant densities for the unperturbed map, which indicates a coherent structure related to the transfer of mass between the two parts of the space. In [10], Dolgopyat and Wright take a similar situation but analyze the restrictions of the map to parts of the space, where the “leaking” of mass is seen as holes in the system. Looking at these open systems, the largest eigenvalues have a particular form related directly to the sizes of the mass transfer/leaking (which are framed as transition probabilities of a related Markov chain). In our case, we are interested in generalizing these ideas to the non-autonomous setting, to see how random mass transfer impacts the value of the second-largest Lyapunov exponent for the cocycle of Perron-Frobenius operators (instead of just a single map).
In the setting of these cocycles of paired tent maps, we are able to show that the hypotheses of Theorem A are true, taking the Banach space to be (L∞ equivalence classes of) bounded variation functions and finding a suitable cone that is preserved by all of the associated Perron-Frobenius operators. Thus we obtain an equivariant density for the cocycle, and an upper bound for the second-largest Lyapunov exponent in terms of 1 and 2 and quantities related to both the map and the cone. Next, we study the
response of the system upon scaling 1 and 2 by some parameter κ and taking κ to
0, which simulates shrinking a perturbation of the map T0,0 back towards the original
map. In this way, we can see how the second-largest Lyapunov exponent behaves under perturbations; one might hope that it shrinks as a nice function of κ (linear, say) until at κ = 0 the top Oseledets space becomes two-dimensional (spanned by the two invariant densities of T0,0) and the zero Lyapunov exponent obtains multiplicity
follow in the body of the paper.
Theorem B. Let (Ω, B, µ, σ) be an ergodic, invertible, probability-preserving transfor-mation, and let 1, 2 : Ω → [0, 1] be measurable functions which are both not µ-a.e.
equal to 0 and which both have countable range. Let Tω = T1(ω),2(ω) be defined as
above. Then there exists a readily computed number C such that λ2 ≤ C < 0 = λ1,
where λ1 and λ2 are the largest and second-largest Lyapunov exponents for the cocycle
of Perron-Frobenius operators associated to Tω(n).
Theorem C. Let (Ω, µ, σ), 1, and 2 be as in Theorem B. Let κ ∈ (0, 1], and consider
the cocycle of maps Tκ(n)
1(ω),κ2(ω). Then there exists c > 0 such that for sufficiently
small κ, the second-largest Lyapunov exponent λ2(κ) for the cocycle of Perron-Frobenius
operators satisfies
λ2(κ) ≤ −cκ.
Theorem D. The estimate in Theorem C is sharp, in the following sense. Set 1 =
2 = 1 for all ω. Then there is a sequence (κn)∞n=1 ⊆ (0, 1/2) such that κn → 0, each
Tκn,κn is Markov, and λ2(κn) is asymptotically equivalent to −2κn.
We emphasize that these results apply to an entire parameterized family of maps, and thus they give a general statement on the asymptotic properties of the second-largest Lyapunov exponent for these maps; to the best of our knowledge, this is the first time λ2 has been upper-bounded for a family of maps, with an asymptotic estimate
on the order of the bound in the scaling parameter. Note also that Theorems B and C are consequences of Theorem A applied to different quantities kP, GP, DP. The
primary work done, outside of showing that the hypotheses of Theorem A are satisfied, is to obtain expressions for each of those quantities. Theorem D is shown by direct computation with a specific class of paired tent maps.
In the process of applying Theorem A to the cocycle of Perron-Frobenius operators associated to the paired tent maps, we happen to require a new Lasota-Yorke-type inequality for Perron-Frobenius operators acting on bounded variation functions. Its utility comes from being sufficiently strong to force small coefficients of the variation terms, but balanced in such a way as to provide uniform bounds on both terms over a family of maps, not just one map individually. The inequality is based on Rychlik’s work [48]; we prove the inequality in a similar level of generality, to provide a tool for future work. For details, see Chapter 3.
The remainder of the dissertation is as follows. In Chapter 2, we give some required background on cones, measurability, and Lyapunov exponents before stating and prov-ing our cocycle Perron-Frobenius theorem. In Chapter 3, we briefly set up, state, and
prove a new balanced Lasota-Yorke-type inequality. In Chapter 4, we use that new Lasota-Yorke inequality to apply our cocycle Perron-Frobenius theorem to cocycles of paired tent maps as described above, to prove the aforementioned bound in Theorem B on the second-largest Lyapunov exponents for the Perron-Frobenius operators, and then find the perturbation estimate in Theorem C. In Chapter 5, we prove Theorem D by studying in-depth a specific class of Markov paired tent maps that turn out to be very amenable to analysis via standard finite-dimensional linear algebra techniques, and allow for explicit computation of the second-largest Lyapunov exponents (through eigenvalues). Theorem D is a step towards trying to answer a related but possibly harder question: what is a lower bound for the second-largest Lyapunov exponent? Finally, there is an appendix containing miscellaneous technical results that are used in various places, and the collection of references at the end.
The majority of this work is contained in two submitted papers; preprints can be found at [23, 24].
Chapter 2
Cocycle Perron-Frobenius Theorem
As mentioned in the Introduction, we can obtain detailed information about invariant densities, ergodicity, and mixing properties of a dynamical system by looking at the spectral properties of the associated transfer operators. In the finite-dimensional dis-crete case of a Markov chain, the operator is simply the transition probability matrix, and we can utilize the classical Perron-Frobenius theorem, proved first by Perron in 1907 for positive matrices [42] and extended by Frobenius to primitive non-negative matrices in 1912 [15].
Let the spectral radius of a matrix A be given by ρ(A). The classical Perron-Frobenius theorem states the following (among other things). Given a primitive non-negative d-by-d matrix P , there exist v, w ∈ (R>0)d and λ ∈ [0, ρ(P )) such that the
following statements are true:
• the spectral radius ρ(P ) is a simple eigenvalue for P , with eigenvector v; • there is no other eigenvalue of P with modulus ρ(P );
• for any x ∈ Rd, we have ρ(P )−nPnx −→
n→∞(w · x)v; and
• if w · x = 0, then kPnxk ≤ Cλn for some C depending on x.
The quantity λ is the modulus of the second-largest eigenvalue(s). When applied to Markov chains, the theorem yields a unique stationary distribution in v (because ρ(P ) is equal to 1), convergence of any initial probability state to that stationary distribution, and the exponential rate of that convergence (λ, as can be seen by writing the vector x−(w·x)v in terms of (generalized) eigenvectors corresponding to the smaller eigenvalues). If λ < ρ(P ), then we say that P has a spectral gap; the spectral gap is the primary driver of the exponential rate of convergence.
We wish to extend the classical Perron-Frobenius theorem as above to a theorem for cocycles of operators on a Banach space that has similar conclusions. The station-ary distribution will be an equivariant vector, and the spectral gap would indicate a
decay rate for all vectors that do not have a component in the direction of the equiv-ariant vector. To do this, we must find an appropriate substitution for the primitivity condition.
As motivation for what the new primitivity condition should be, observe that if x is a vector with non-negative entries, then a non-negative matrix P preserves that property: P x also has non-negative entries. Said another way, the matrix P preserves the non-negative orthant Rd≥0. This fact is a very specific case of the much more general
principle of positive operators, which is general is described using cones. We will give background on positivity and a related quantity called the projective metric. After describing measurable and topological considerations, as well as recapping facts about cocycles and Lyapunov exponents, we will state and prove our generalization of the Perron-Frobenius theorem, Theorem 2.4.3. Afterwards, we apply the theorem to a number of easy examples to demonstrate its strength and limitations.
2.1
Cones
The generalization of positivity from R to general vector spaces is through cones. We will define the specific type of cones we need on Banach spaces and provide relevant examples to our applications. From the definition of our cones we obtain a partial order and a projective pseudo-metric on the cones, both of which we relate to the norm on the Banach space. The relationship between the cone, the pseudo-metric, and the norm provides sufficient structure to understand the implications of bounded linear operators that preserve the cone.
Cone Properties and Examples
We begin with the definitions and immediate results. Following that, we provide exam-ples that will be used elsewhere. References for some of this material include Schaefer’s classic text on topological vector spaces [49] and Peressini’s book on ordered topological vector spaces [41].
Definition 2.1.1. Let (X, k·k) be a real Banach space. A cone is a set C ⊆ X that is closed under scalar multiplication by positive numbers, i.e. λC ⊆ C for all λ > 0. We define a nice cone on (X, k·k) to be a cone C ⊆ X that has the following properties:
• C is convex (equivalently, closed under addition);
• C is blunt, i.e. 0 /∈ C (as opposed to pointed, where 0 ∈ C); • C is salient, i.e. C ∩ (−C) = ∅ (more generally ⊆ {0});
• C is generating (or generates X), i.e. C − C = X;
• C is D-adapted (to k·k), i.e. there exists D ∈ R≥1 such that for x ∈ X and y ∈ C,
if y ± x ∈ C ∪ {0}, then kxk ≤ D kyk. If x ∈ C, we say that x is a positive element of X.
The terminology is mostly self-explanatory, with the exception of “salient”: this word can refer to something pointing outward, roughly speaking. A cone that contains no vectors also in its negative (that is, a cone that contains no one-dimensional sub-space) can be seen as identifying an outward direction, so the term salient is not as strange as it first appears.
From the cone, we obtain a partial order on the Banach space; this order justifies calling elements of the cone “positive”. Alternatively, from any partial order on the Banach space we can take the positive elements to form a salient convex cone, but we will not need this perspective.
Lemma 2.1.2. Let X be a real Banach space, and C a salient convex cone. Then C induces a partial order on X, denoted C (or when the choice of C is clear), by
x C y when y − x ∈ C ∪ {0}. Moreover, is a vector order; that is, if c > 0,
x, y, z ∈ X, and x y, then cx cy and x + z y + z. Finally, if C ∪ {0} is closed, then whenever xn 0 for all n and xn converges to x, we have x 0 also.
Proof. If x ∈ X, then x C x because x − x = 0 ∈ C ∪ {0}. If x C y and y C x, then
both y − x ∈ C ∪ {0} ∩ − C ∪ {0} = {0}, by salience of the cone, so x = y. If x C y
and y C z, then we have
z − x = (z − y) + (y − x) ∈ C ∪ {0}, so that x C z.
Suppose that c > 0, x, y, z ∈ X, and x y. Then c(y − x) ∈ C ∪ {0} and (y + z) − (x + z) = y − x ∈ C ∪ {0}, so cx cy and x + z y + z. If C ∪ {0} is closed, the limit of any sequence in C ∪ {0} remains in C ∪ {0}, which proves the statement. Remark 2.1.3. When Lemma 2.1.2 applies, we see that the D-adapted condition can be rephrased to say that if −y C x C y, then kxk ≤ D kyk. Note that this inequality
forces y ∈ C. The D-adapted condition is, therefore, a way of connecting the order and the norm. With it, if an element of the space is bounded in the order, then it is actually bounded in the norm as well; this fact will be used to great benefit later.
In the literature, if there is some D ≥ 1 such that C is D-adapted in (X, k·k), then C is often called normal and there exists an equivalent norm to k·k such that C is 1-adapted with respect to that norm [49, Section V.3]. We will not use this fact, opting instead to work using the existing norm (the equivalent norm is a Minkowski functional for an appropriate saturated convex set).
Lemma 2.1.4. Let (X, k·k) be a real Banach space with a salient closed convex cone C. If C has non-empty interior (with respect to k·k) or if is a lattice order (for a pair of vectors x, y ∈ X, there exists z ∈ X, denoted z = x ∨ y, such that (x + C) ∩ (y + C) = z + C), then C generates X.
For a lattice order, the quantity x ∨ y is also denoted sup(x, y), since it is the least upper bound for x and y.
Proof. Suppose that z an interior point of C. If x ∈ X and > 0 such that z + x ∈ C, observe that x = 1 (z + x) − 1 z ∈ C − C. Hence C generates X.
Now suppose that is a lattice order for X. Define x+= x ∨ 0 and x− = (−x) ∨ 0; we claim that x = x+−x−. Observe that for any z ∈ X, z+x∨y ∈ (x+z+C)∩(y+z+C),
and any w ∈ (x+z+C)∩(y+z+C) is at least z+x∨y in ; thus z+x∨y = (x+z)∨(y+z). By this equation and the definitions of x±, we have
x + x− = (x − x) ∨ (x + 0) = x+,
and hence x = x+− x−. Thus C generates X (potentially by adding and subtracting
a non-zero cone vector).
Example 2.1.5. Let X = Rd and equip X with the 1-norm k·k1. Then C = Rd≥0\ {0} is a nice cone called the positive orthant. The sum of two non-negative real numbers is non-negative, so C is closed under addition; by definition C is blunt and salient, and the closure of C is C ∪ {0}. To see that C is generating, observe that is a lattice order on Rdand apply Lemma 2.1.4 (in particular, x± are defined by (x+)i = max{0, xi} and
(x−)i = max{0, −xi}).
Note that x y if and only if xi ≤ yi for each i = 1, . . . , d. To see that C is
D-adapted, suppose that −y x y; from this inequality, we have 0 2yi for each
i, so each yi is non-negative. Then −yi ≤ xi ≤ yi for all i, or |xi| ≤ yi. Compute the
norms: kxk1 = d X i=1 |xi| ≤ d X i=1 yi = kyk1. Thus C is 1-adapted to k·k1.
Example 2.1.6. Let (X, k·k) be a real Banach space, let f ∈ X∗be a non-zero bounded linear functional, and let α ∈ (0, kf k). Then the subset Cf,α ⊆ X defined by
is a kf kα -adapted blunt salient closed convex cone. To check that Cf,α satisfies the
conditions, we use the properties of f as a bounded linear functional.
Suppose that x ∈ Cf,α and c > 0. We have f (cx) = cf (x) ≥ cα kxk = α kcxk, so
cx ∈ Cf,α, and Cf,α is a cone. If x, y ∈ Cf,α and λ ∈ (0, 1), we have
f (λx + (1 − λ)y) = λf (x) + (1 − λ)f (y) ≥ α(λ kxk + (1 − λ) kyk) ≥ α kλx + (1 − λ)yk , hence Cf,α is convex. By definition, Cf,α is blunt, as it does not contain 0. Suppose
that x ∈ Cf,α∩ −Cf,α. Then f (x) ≥ α kxk ≥ 0, but −x ∈ Cf,α, and so
0 ≥ −f (x) = f (−x) ≥ α k−xk = α kxk ,
which implies that x = 0, a contradiction. Thus Cf,α∩−Cf,αis empty, i.e. Cf,α is salient.
If xn −→
n→∞x 6= 0 and each xn∈ Cf,α, then observe that by continuity of the norm,
f (x) = lim
n→∞f (xn) ≥ limn→∞α kxnk = α kxk .
Hence x ∈ Cf,α, and Cf,α∪ {0} is closed.
To see the fact that Cf,α is kf k
α -adapted, suppose that −y x y for x, y ∈ X.
Then f (y) ≥ α kyk ≥ 0, since y ∈ Cf,α∪ {0}. Moreover, we have y ± x ∈ Cf,α∪ {0}, so
that
f (y) + f (x) = f (y + x) ≥ α kx + yk , f (y) − f (x) = f (y − x) ≥ α ky − xk . We then apply the triangle inequality to obtain:
kxk ≤ 1 2 ky + xk + ky − xk ≤ 1 2α f (y) + f (x) + f (y) − f (x) = f (y) α ≤ kf k α kyk , so that Cf,α is kf k α -adapted.
Finally, suppose that there exists x ∈ Cf,α such that f (x) > α kxk. We claim that
x is an interior point of Cf,α, so that Cf,α generates X by Lemma 2.1.4. Let > 0 be
less than (f (x) − α kxk)(α + kf k)−1 and let z ∈ B(0, ). Then we have α kzk − f (z) ≤ |α kzk − f (z)| ≤ (α + kf k) kzk
|{z}
<
< f (x) − α kxk ,
which then implies
f (x + z) > α(kxk + kzk) ≥ α kx + zk .
Bishop-Phelps cone [43, 51].
Note also that in this situation, f is a strictly positive linear functional; not only is f non-negative on C, it sends every element of C to a positive number, using the fact that f (x) ≥ α kxk > 0 for all x ∈ C (as C is blunt). Strictly positive linear functionals have interesting properties regarding cones; their existence is equivalent to existence of a convex “base” for the cone (see [41]), and if a linear operator preserves a cone, then the adjoint of that operator sends strictly positive linear functionals (with respect to that cone) to strictly positive linear functionals. Moreover, positive linear functionals are automatically bounded in some cases (depending on the properties of the cone). We will not have need to use any of this going forward, because we need to compute uniform bounds on the norm of the linear functionals we handle, but it should be noted for the purposes of being aware of other abstract theory.
Example 2.1.7. If X is a (real) Hilbert space, y ∈ X is non-zero, and α ∈ (0, kyk), then because y represents the linear functional x 7→ hx, yi, we have that
Cy,α = {x ∈ X : hx, yi ≥ α kxk} \ {0}
is a nice cone in X. We make the following claim: If c > 0, then (cy + {y}⊥) ∩ Cy,α = (cy + {y}⊥) ∩ B
cy,c kyk α q kyk2− α2 .
To see this equality, let x ∈ cy + {y}⊥, and note that hx, yi = c kyk2. Then, we have
kx − cyk2 = kxk2− 2chx, yi + c2kyk2 ≤ c2kyk 4 α2 − c 2kyk2 if and only if (α kxk)2 ≤ c2kyk4 − 2c2α2kyk2 + 2cα2hx, yi = (c kyk2)2 = hx, yi2,
which is if and only if α kxk ≤ hx, yi, because both quantities are non-negative. This equality indicates that the perpendicular plane to cy intersects Cy,α in a disk-shaped
region, and so the cone looks very much like an infinite-dimensional “ice cream” cone (think of waffle cones). The closer α gets to 0, the larger the disk becomes, approaching a half-plane; the closer α gets to kyk, the smaller the disk becomes, approaching the positive ray through y. In this picture, the quantity D is something like an aperture for the cone. The same idea holds for general Cf,α, though without the disk shape, and
noting that if we tried to set α = kf k, Cf,α could be empty (there exist bounded linear
functions that do not achieve their norm; we avoid this case entirely in our definition). Informally, we might call all Cf,α “ice cream” cones, because of the Hilbert space case.
y
1
2y + {y} ⊥
Figure 2.1: The cone Cy,α for y = (1, 2) and α = 2 in R2,
depicted with the perpendicular plane at cy with c = 1/2.
The proofs of all of the statements in the following lemma are simply appeals to the definitions of the various concepts.
Lemma 2.1.8. If C1 and C2 are two cones in a real Banach space (X, k·k), then C1∩ C2
is also a cone; if both C1 and C2 are closed or convex or blunt, then C1∩ C2 is closed or
convex or blunt, respectively. If at least one of C1 or C2 is salient or D-adapted (say,
constants D1 and D2, potentially one of which being infinity), then C1 ∩ C2 is salient
or D-adapted with D = min{D1, D2}, respectively. If x ∈ C1∩ C2 is an interior point
to both C1 and C2, then it is an interior point to C1∩ C2.
Example 2.1.9. Consider the Banach space C0(X), where X is a non-compact locally
compact Hausdorff topological space (for example, X = Z≥0 equipped with the discrete
topology), equipped with the supremum norm. Let C ⊆ C0(X) be the cone of positive
continuous functions, C = {f ∈ C0(X) : f ≥ 0 and f (x) > 0 for some x}. We will
show that C is a nice cone with empty interior. By definition 0 /∈ C, and positive scalar multiples of functions in C remain in C. If f1, f2 ∈ C and λ ∈ (0, 1), then if f1(x) > 0,
we see (λf1+ (1 − λ)f2)(x) > 0, so C is convex. The intersection of C and −C is empty,
as any such elements of the intersection would be both non-negative and non-positive, hence zero; thus C is salient. We see that C ∪ {0} is closed: if fn converges to f in
C0(X), then f takes all non-negative values. To see that C generates C0(X), observe
that is a lattice order: if f, g ∈ X, then the pointwise max(f, g) is also continuous and is equal to f ∨ g (any function larger than both f and g is at least max(f, g)). By Lemma 2.1.4, C generates C0(X). Finally, if −g f g, then for all x ∈ X,
|f (x)| ≤ g(x), and so kf k∞ ≤ kgk∞. Thus C is 1-adapted to .
However, C has empty interior. Let f ∈ C, let > 0, and find x ∈ X such that 0 ≤ f (x) < /2. Then Bk·k∞(f, ) is not contained in C, since we can construct a continuous
function h ∈ B(0, ) ⊆ C0(X) with h(x) > /2 and observe that f − h ∈ B(f, ) but
f (x) − h(x) < 0, hence f − h /∈ C.
Note that essentially the same arguments show that the cone of positive functions in l1(Z
Example 2.1.10. Let Lip[0, 1] be the Banach space of Lipschitz functions on [0, 1], equipped with the norm
kf kLip= max{kf k∞, Lip(f )}, and for a > 0, let
Ca = f ∈ Lip[0, 1] : f (x) > 0 ∀x ∈ [0, 1],f (x) f (y) ≤ e a|x−y| ∀x, y ∈ [0, 1] . Then Ca is a nice cone on Lip[0, 1], with D-adapted constant D = 1 + 2aea; Liverani
uses this cone in [33].
Let f ∈ Ca, and let c > 0. Then
cf (x) cf (y) = f (x) f (y) ≤ e a|x−y|, so cf ∈ C a. If f, g ∈ Ca,
then by definition f + g takes positive values, and moreover for all x, y ∈ [0, 1] we have f (x) − f (y)ea|x−y| ≤ 0 and g(y)ea|x−y|− g(x) ≥ 0. This implies
f (x) − f (y)ea|x−y| ≤ 0 ≤ g(y)ea|x−y|− g(x),
and rearranging the resulting inequality yields f (x) + g(x) f (y) + g(y) ≤ e
a|x−y|,
which means f + g ∈ Ca. These two conditions imply that Ca is convex. Of course,
0 /∈ Ca, and the intersection Ca∩ −Ca is empty.
To see that Ca generates Lip[0, 1], observe that 1 ∈ Lip[0, 1]. We will show that 1
is an interior point of Ca and apply Lemma 2.1.4. Observe that for f ∈ B(0, 1) and
x 6= y ∈ [0, 1], we have: 1 + f (x) 1 + f (y) = 1 + f (x) − f (y) 1 + f (y) ≤ 1 + Lip(f ) |x − y| 1 − kf k∞ ≤ 1 + kf kLip|x − y| 1 − kf kLip ≤ exp kf kLip 1 − kf kLip|x − y| ! ,
where we use the power series expansion of et. For kf k
Lip ≤ a(1 + a)
−1, we have that
kf kLip(1 − kf kLip)−1 ≤ a, and so 1 + f ∈ Ca. Hence B(1, a(1 + a)−1) ⊆ Ca.
Then, if fn ∈ Ca and fn −→
n→∞ f in Lip[0, 1], then either f ∼= 0, or f is never 0. To
see this, suppose that f (x) = 0 for some x ∈ [0, 1]. Then for any y ∈ [0, 1] we have f (y) − f (x)ea|y−x|= lim
n→∞fn(y) − fn(x)e
a|y−x| ≤ 0,
uniformly 0. In the case where f is never zero, the same limit calculation shows that f (y) f (x) ≤ e a|y−x| , and so f ∈ Ca.
Finally, suppose that f, g ∈ Lip[0, 1], with −f g f . Then f (x) − g(x) > 0 and g(x) + f (x) > 0 for all x ∈ [0, 1], implying that f (x) > 0 and |g(x)| < f (x). This yields kgk∞≤ kf k∞. Then, for x, y ∈ [0, 1], we have
e−a|x−y| ≤ f (x) − g(x) f (y) − g(y) ≤ e
a|x−y|
,
by both sides of the order relation −f g f . We then subtract 1 to see that f (x) − g(x)
f (y) − g(y) − 1 ∈e
−a|x−y|− 1, ea|x−y|− 1 .
Since we also have
e−a|x−y|− 1
= 1 − e
−a|x−y|
= e−a|x−y|(ea|x−y|− 1) ≤ ea|x−y|− 1,
we obtain f (x) − g(x) f (y) − g(y) − 1 ≤ ea|x−y|− 1
for all x, y ∈ [0, 1]. Then we have:
|g(x) − g(y)| = |g(x) − f (x) + f (x) − f (y) + f (y) − g(y)| ≤ |f (x) − f (y)| + |(g(x) − f (x)) − (g(y) − f (y))| ≤ Lip(f ) |x − y| + |f (y) − g(y)| ·
f (x) − g(x) f (y) − g(y) − 1 ≤ Lip(f ) |x − y| + 2 |f (y)| · ea|x−y|− 1 .
By the Mean Value Theorem, if x 6= y then we have ea|x−y|− 1 = |x − y| · aeat for some
t ∈ (0, |x − y|). Using the fact that this t is at most 1, we obtain: |g(x) − g(y)| ≤ kf kLip|x − y| + 2 kf k∞|x − y| aea
= kf kLip|x − y| (1 + 2aea) .
Thus Lip(g) ≤ (1 + 2aea) kf k
Lip, and so we have:
Thus Ca is (1 + 2aea)-adapted to k·kLip, as desired.
A different computation, one with power series, can show that Cais (2ea−1)-adapted
to k·kLip; this other constant is smaller when a is larger than 1.
The Projective (Pseudo-)Metric
Given a cone on a real Banach space that generates a partial order, we can define some-thing like a distance on the cone referring only to the order structure. This quantity is actually a projective pseudo-metric: it captures information about something like an angle between two vectors in the cone, so collinear vectors are distance zero, but near the boundary of the cone the distance grows very large, and the boundary components are infinitely far from other parts of the cone. Moreover, we will see shortly that if the cone is nice, then the distance is closely related to the norm; this relation will give us a way to translate results about operators and cones into analytic properties. Here, we make the requisite definitions and then show that the quantities involved have sufficiently nice properties for use going forward.
Definition 2.1.11. Let (X, k·k , C) be a real Banach space with a salient closed convex cone. For v, w ∈ C, define:
α(v, w) = sup {λ ≥ 0 : λv w} ; β(v, w) = inf {µ ≥ 0 : w µv} ; θ(v, w) = log β(v, w) α(v, w) .
The quantity θ(v, w) is the projective (pseudo-)metric on C, and is sometimes called a Hilbert metric on C. If f, g ∈ C have θ(f, g) < ∞, then we say that f and g are comparable.
Recall that v w by definition means that w − v ∈ C ∪ {0}; we will use both notations interchangeably as required. We gather important properties about α and β in Proposition 2.1.12 and properties of θ in Proposition 2.1.13.
Proposition 2.1.12. Let (X, k·k , C) be a real Banach space with a salient closed convex cone.
• In the second component, α is super-additive, β is sub-additive, and both are positive-scalar-homogeneous, i.e. for all v, w, z ∈ C and c > 0:
α(v, w + z) ≥ α(v, w) + α(v, z), β(v, w + z) ≤ β(v, w) + β(v, z), α(v, cw) = cα(v, w), β(v, cw) = cβ(v, w).
• The quantities α and β are increasing in the second component: if v, w, z ∈ C and w z, then
α(v, w) ≤ α(v, z), β(v, w) ≤ β(v, z).
• The quantities α and β have a symmetry property: for all v, w ∈ C, α(v, w) = β(w, v)−1.
• If v, w ∈ C are comparable, then α(v, w) and α(w, v) > 0, and β(v, w) and β(w, v) < ∞.
• For all v, w ∈ C, we have α(v, w) ≤ β(v, w). • For all v ∈ C, α(v, v) = 1 = β(v, v).
If C is, moreover, D-adapted, then for all v, w ∈ C we have α(v, w) ≤ Dkwk kvk, β(v, w) ≥ 1 D kvk kwk.
Finally, if L : X → X is a linear operator such that LC ⊆ C, then for all v, w ∈ C: α(L(v), L(w)) ≥ α(v, w), β(L(v), L(w)) ≤ β(v, w).
Proof. Note that C ∪ {0} is closed. This fact implies that w − α(v, w)v ∈ C ∪ {0}, because w − λv ∈ C ∪ {0} for all λ < α(v, w) and w − α(v, w)v is a limit point of these w − λv. We can thus write:
w + z − α(v, w) + α(v, z)v = w − α(v, w)v + z − α(v, z)v ∈ C ∪ {0},
so that α(v, w + z) ≥ α(v, w) + α(v, z). The analogous proof holds for β(v, w)v − w ∈ C ∪ {0} and the β sub-additivity. If c > 0, then we have
α(v, cw) = sup {λ > 0 : λv cw} = c · sup λ c > 0 : λ cv w = cα(v, w), and similarly for β.
Suppose that v, w, z ∈ C and w z. Then
z − α(v, w)v = z − w + w − α(v, w)v ∈ C ∪ {0}, so that α(v, z) ≥ α(v, w). Similarly,
