University of Zagreb Faculty of Science Department of Mathematics
Fixed points of diffeomorphisms, singularities of vector fields and epsilon-neighborhoods of their orbits
Pavao Mardešić, Université de Bourgogne, Dijon, France
Vesna Županović, University of Zagreb, Zagreb, Croatia
In the first place, I would like to thank my two supervisors, Professor Pavao Mardešić and Professor Vesna Županović, for making this work possible.
I am enormously grateful to Professor Pavao Mardešić, for proposing the subject of my thesis, for inviting me to spend my final year at the University of Burgundy to work with him intensively, for always having time and patience to discuss with me, without exception. I am thankful for all the ideas and motivating talks, which kept me interested and moving forward all the time, for the optimism and curiosity he passed along to me, as well as for all the encouragement and support I received.
I am extremely thankful to Professor Vesna Županović for many useful ideas, valuable advices and great help and support, for taking care of my progress from the very beginning, and always providing fresh ideas and motivation for research in the field of fractal geometry. I am grateful for our numerous mathematical discussions which kept my work in progress during the last six years.
I would also like to thank all the members of the Seminar for differential equations and nonlinear analysis in Zagreb for motivating talks in fractal analysis. Special thanks goes to Professor Darko Žubrinić, for useful discussions, and to Professor Mervan Pašić and Professor Luka Korkut. I owe my thanks also to many professors I have met during my stay abroad, for their willingness to hear about my research and discuss with me: Professors Jean-Philippe Rolin, David Sauzin, Frank Loray, Guy Casale, Loïc Teyssier, Reynhard Schaefke and Johann Genzmer.
Finally, I thank the commitee members for my thesis defense, Professor Jean-Philippe Rolin, Professor Siniša Slijepčević, Professor Sonja Štimac and Professor Darko Žubrinić.
In the end, I would like to mention the financial support that enabled me to spend my final year in Dijon and to finish my thesis: the French government scholarship for the academic year 2012/13 and the 2012 Elsevier/AFFDU grant.
In the thesis, we consider discrete dynamical systems generated by local diffeomorphisms in a neighborhood of an isolated fixed point. Such discrete dynamical systems associate to each point its orbit. We investigate to what extent we can recognize a diffeomorphism and read its intrinsic properties from fractal properties of one of its orbits. By fractal properties of a set, we mean its box dimension and Minkowski content. The definitions of the box dimension and the Minkowski content of a set are closely related to the notion of -neighborhoods of the set. More precisely, one considers the asymptotic behavior of the Lebesgue measure of the -neighborhoods, as goes to zero. Thus, in a broader sense, by fractal property of a set we mean the measure of the -neighborhood of the set, as a function of small parameter . When necessary, in the thesis we introduce natural generalizations of fractal properties, which are better adapted to our problems. The relevance of this method lies in the fact that fractal properties of one orbit can be computed numerically.
Local diffeomorphisms appear naturally in many problems in dynamical systems. For example, they appear in planar polynomial systems as the first return maps or Poincaré maps of spiral trajectories, defined on transversals to monodromic limit periodic sets (singular elliptic points, periodic orbits and hyperbolic polycycles). With appropriate parametrizations of the transversal, Poincaré maps are local diffeomorphisms on , except possibly at zero, having zero as an isolated fixed point. Recognizing the multiplicity of zero as a fixed point of the first return map is important for determining an upper bound on the number of limit cycles that are born from limit periodic sets in unfoldings. This number of limit cycles is called the cyclicity of the limit periodic set. The problem is closely related to the Hilbert problem.
In the first part of this thesis, we show that there exists a bijective correspondence between the multiplicity of zero as a fixed point of a diffeomorphism and the appropriate generalization of the box dimension of its orbit, which we call the critical Minkowski order.
Another occurrence of diffeomorphisms is when considering the holonomy maps in . In particular, we consider germs of complex saddle vector fields in . Their holonomy maps on cross-sections transversal to the saddle are germs of diffeomorphisms , with isolated fixed point at the origin. It is known that the formal and the analytic normal form of such fields can be deduced from the formal and the analytic classes of their holonomy maps.
In the second part of the thesis, we therefore study germs of diffeomorphisms , from the viewpoint of fractal geometry. We restrict ourselves to germs whose linear part is a contraction, a dilatation (the so-called hyperbolic fixed point cases) or a rational rotation (the so-called parabolic fixed point cases). The hyperbolic cases are easy to treat and most of the time we deal with the non-hyperbolic case of rational rotations, that is, with parabolic germs. In this work we omit the very complicated case when the linear part is an irrational rotation.
We give the complete formal classification result: there exists a bijective correspondence between the formal invariants of a germ of a diffeomorphism and the fractal properties of any of its orbits near the origin. We use generalizations of fractal properties that we call directed fractal properties.
On the other hand, we have not solved the analytic classification problem for parabolic germs, using fractal properties of orbits. We state negative results in this direction and problems that are encountered. For germs inside the simplest formal class, we investigate analytic properties of -neighborhoods of orbits and compare them with the well-known results about analytic classification.
Keywords: -neighborhoods, box dimension, Minkowski content, fixed points, germs of diffeomorphisms, multiplicity, Poincaré map, cyclicity, parabolic germs, complex saddle vector fields, holonomy maps, saddle loop, formal classification, analytic classification, Abel equation, Stokes phenomenon
- 0 Introduction
1 Application of fractal analyis in reading multiplicity of fixed points for diffeomorphisms on the real line
- 1.1 Introduction and definitions
- 1.2 Generators differentiable at a fixed point
- 1.3 Non-differentiable generators at a fixed point
- 1.4 Application to cyclicity for planar vector fields using fractal analysis of Poincaré maps
- 1.5 Application to number of zeros of Abelian integrals
2 Application of fractal analysis in formal classification of complex diffeomorphisms and saddles
- 2.1 Introduction
- 2.2 Hyperbolic germs
- 2.3 Formal classification of parabolic germs
- 2.4 Application of the formal classification result to formal orbital classification of complex saddles in
3 About analytic classification of complex parabolic diffeomorphisms using -neighborhoods of orbits
- 3.1 Ecalle-Voronin moduli of analytic classification
- 3.2 Analyticity of solutions of generalized Abel equations
- 3.3 Analyticity properties of complex measures of -neighborhoods of orbits of parabolic germs
- 3.4 Principal parts of complex measures of -neighborhoods of orbits and analytic classification
- 3.5 Perspectives
- 3.6 Proofs of auxiliary results
- 4 Conclusions and perspectives
Chapter 0 Introduction
In this thesis, we use the methods of fractal analysis. Our main tool is computing the box dimension and the Minkowski content of sets. The box dimension of trajectories of a dynamical system is an appropriate tool for measuring complexity of the system, and it reveals important properties of the system itself. By their definition, box dimension and Minkowski content of sets are related to the first term in the asymptotic development of the Lebesgue measure of the -neighborhood of the set, as tends to zero, see precise definitions in Section 0.3. In our considerations, we sometimes mean -neighborhoods of sets, for small parameters , as their fractal property in the broader sense. We mainly consider discrete dynamical systems generated by diffeomorphisms in a neighborhood of a fixed point and tending to the fixed point. We conclude intrinsic properties of the generating diffeomorphism by fractal analysis of one of its orbits. The properties we read are important in light of the Hilbert problem for planar polynomial vector fields. The fractal method for obtaining them used in this work is new. We investigate where are its limits in recognizing the diffeomorphism. The applicability of our method lies in the fact that fractal properties of one orbit are of purely geometric nature and can be determined numerically.
After a few words about fractal analysis and its historical development, we describe how it was exploited so far in the field of dynamical systems. Fractal analysis has been rapidly evolving since the end of the century. It was noted that many sets in the nature (for example, coastline or a snow-flake) have fractal structure, meaning that at every point they are of infinite length and have self-similarity property: any part is similar to the whole. The need to measure their complexity led to the introduction of new notion of fractal dimension, different from topological dimension, which takes noninteger values, depending on the complexity of the set. The motivation can be found in e.g. Mandelbrot’s book  about fractals in nature. The famous fractal sets are Koch curve, Cantor set, Sierpinski carpet or Julia set. Their fractal dimensions can be found in e.g. .
The oldest and most widely used fractal dimension is Hausdorff dimension, introduced by Carathéodory. There are many works where Hausdorff dimension was exploited in dynamical systems. For example, in papers of Douady, Sentenac, Zinsmeister  and Zinsmeister , the discontinuity in Hausdorff dimension at a parabolic fixed point of Julia set indicates the moment of change of local dynamics (the so-called parabolic implosion phenomenon). Many dynamical systems posess the strange, chaotic attractors with fractal structure, for example Lorenz attractor, Smale horseshoe, Hénon attractor. They are difficult to describe and Hausdorff dimension shows the complexity of such attractors. For their fractal dimension and its application in dynamical systems, see for example the overview article of Županović, Žubrinić  and references therein.
The notion of box dimension was introduced later, at the beginning of the century, by Minkowski and Bouligand. In literature, it is also called the limit capacity or the box counting dimension. For the overview and more information on fractal dimensions, see for example the book of Falconer  or the book of Tricot .
In this work we use the box dimension and the Minkowski content as relevant fractal properties of sets. The extensive use of box dimension in the study of dynamical systems and differential equations started around the year 2000 by a group of authors in Zagreb: Pašić, Žubrinić, Županović, Korkut et al., in papers e.g. , , , . The use of box dimension in their work was motivated by the book of Tricot , which provides box dimension of two special sets: of the graph of -chirp function: , , , and of spiral accumulating at the origin: , . Also, in , Lapidus and Pomerance computed the box dimension of one-dimensional discrete sequences accumulating at zero with well-defined asymptotics and connected it in their modified Weyl-Berry conjecture to the asymptotics of eigenvalue counting function for fractal strings.
Box dimension of sets in takes values in the interval . It takes also noninteger values, depending on how much of the ambient space the set occupies (the density of the accumulation of the set). The precise definitions are given in Section 0.3. For most sets, it coincides with Hausdorff dimension. Nevertheless, the Hausdorff dimension, unlike the box dimension, posesses the countable stability property, see e.g. . This results in the fact that any countable set of poins accumulating at some point, regardless of the density of the accumulation, has Hausdorff dimension equal to . Similarly, any countable set of smooth curves accumulating at some set has Hausdorff dimension equal to . On the other hand, their box dimension takes values between and , or and respectively, depending on the density of the accumulation. To conclude, Hausdorff dimension for the trajectories of continuous and discrete dynamical systems is trivial and provides no information. The box dimension is an appropriate tool.
This thesis is a natural continuation of the work of the above mentioned group of professors in Zagreb. The considerations are extended to systems where standard box dimension is not enough and we need natural generalizations. Here we present shortly an overview of relevant former results and explain what is done in the thesis.
In a broader sense, the results of Žubrinić and Županović are mostly concerned with the Hilbert problem, from the viewpoint of fractal geometry. Planar polynomial vector fields are considered. The Hilbert problem asks for an upper bound , depending only on the degree of the polynomial field, on the number of limit cycles (isolated periodic orbits). The problem is so far completely open. To approach this question, it is important to detect invariant sets from which limit cycles are born in generic analytic unfoldings of vector fields. They are called limit periodic sets. The question then reduces to a simpler question, on the maximal number of limit cycles that can be born from each limit periodic set in a generic analytic unfolding. This is called the cyclicity of the limit periodic set. A nice overview of the Hilbert problem and related problems can be found in the book of Roussarie .
We restrict ourselves to the simplest cases of monodromic limit periodic sets: isolated singular elliptic points (singularities with eigenvalues with nonzero imaginary part, i.e., strong or weak foci), limit cycles, and homoclinic loops (with a hyperbolic saddle point at the origin). Monodromic means that the set is accumulated on one side by spiral trajectories. In these cases, the cyclicity is finite (Dulac problem proven by Il’yashenko  and Ecalle  independently) and known, as is also in some special cases of polycycles. See works of Mourtada, El Morsalani, Il’yashenko, Yakovenko and others, as referenced in [46, Chapter 5].
The question arose if it was possible to read these bounds using this new method: fractal analysis of spiral trajectories tending to the limit periodic set. The idea was triggered by article  of Žubrinić and Županović. In the article, the example of Hopf bifurcation was treated from the viewpoint of fractal analysis. In Hopf bifurcation, a weak focus of the first order bifurcates to strong focus and one limit cycle is born at the moment of bifurcation. It was noted that the box dimension of a spiral trajectory around the focus point jumps from the value for weak focus to the smaller value at the very moment of bifurcation, obviously signaling the birth of a limit cycle from the weak focus point. To generalize the result to all focus points, in the next article  by the same authors, the Takens normal form for a generic unfolding of a vector field in a neighborhood of a focus point from Takens  was related to the box dimension of spiral trajectories tending to the focus point. Thus the box dimension of a spiral trajectory accumulating at a focus point recognizes between strong and weak foci and, further, between weak foci of different orders. It thus signals the number of limit cycles that are born from them in perturbations. In , box dimension of spiral trajectories was related to cyclicity also for limit cycles. In computing the box dimension of a spiral trajectory, the box dimension of an orbit of the Poincaré map on a transversal was computed, and related to the box dimension of a spiral trajectory by the flow-sector (for focus points) or the flow-box theorem from (for limit cycles). The theorems state a product structure of a trajectory locally around a transversal. This suggested that the box dimension of a spiral trajectory around limit periodic sets in fact carries the same information as the box dimension of a discrete, one-dimensional orbit of its Poincaré map. The box dimension of (relevant) one-dimensional discrete systems was computed in Elezović, Županović, Žubrinić . Later, in her thesis  and in paper , Horvat-Dmitrović considered bifurcations of one-dimensional discrete dynamical systems, noting that a jump in the box dimension of the system indicates the moment of bifurcation, while its size reveals the complexity of bifurcation. The connection with Poincaré maps for continuous planar systems was stressed as an application, the birth of limit cycles in the unfolding corresponding to the bifurcation of a fixed point of the Poincaré map into new fixed points. Indeed, the cyclicity in generic unfoldings of weak foci and limit cycles equals the multiplicity of zero as a fixed point of the corresponding Poincaré map. Instead of considering the spiral trajectories, one can equivalently perform fractal analysis of one-dimensional orbits of the Poincare map on a transversal to get information on cyclicity. In , one-dimensional discrete systems generated by functions sufficiently differentiable at a fixed point were considered. The bijective correspondence was found between the box dimension of such systems and the multiplicity of fixed points of the generating functions.
In the first part of this thesis, we express an explicit bijective correspondence between the cyclicity of elliptic points and limit cycles in generic unfoldings and the behavior of the -neighborhood of any orbit of their Poincaré maps, as . The behavior is expressed by the box dimension. Then we generalize the results to homoclinic loops and simple saddle polycycles. The results were published in 2012 in the paper  by Mardešić, Resman, Županović. Unlike in focus or limit cycle case, where Poincaré map was differentiable at fixed point and could be expanded in power series, the first return map for homoclinic loop is no more differentiable at fixed point. The theorem from  connecting the multiplicity of differentiable generators with the box dimension of their orbits cannot be applied. However, it is known that the Poincaré maps for generic analytic unfoldings of a homoclinic loop have nice structure: they decompose in a well-ordered (by flatness at zero) scale of logarithmic monomials, which mimics in some way the power scale. This result is given in book of Roussarie [46, Chapter 5]. Such scale is an easy example of the so-called Chebyshev scale. The Chebyshev scales are discussed in detail in the book of Mardešić . We encounter two problems in generalizing the previous result to non-differentiable generators belonging to a Chebyshev scale. First, the multiplicity, to be well defined, should be exchanged with multiplicity in a Chebyshev scale. A definition was introduced by Joyal, . Secondly, we show that for generators not belonging to the power scale, box dimension of their discrete orbits is not precise enough to reveal the exact multiplicity of the generator. By its very definition, box dimension is adapted to the power scale: it compares the area of the -neighborhood of sets with powers of . Even Tricot in his book [53, p. 121] warns about lack of precision of box dimension for sets with logarithmic dependence on of the area of their -neighborhood and emphasizes the need for appropriate, finer scale to which this area should be compared. We introduce a new notion of critical Minkowski order, which presents a generalization of box dimension which is adapted to a Chebyshev scale. It compares the behavior of the -neighborhoods with appropriate, finer scale for a given problem. With this new notion, we manage to recover a bijective correspondence as before. In cases of homoclinic loops, knowing the critical Minkowski order of only one orbit of Poincaré map, together with understanding the scale for a generic unfolding, are sufficient to determine the cyclicity of the homoclinic loop. We stress that the problem of our method for more complicated saddle polycycles lies in the fact that the depth of the logarithmic scale for generic unfoldings is not known in general. The scales have been investigated only in very special cases of polycycles, by El Morsalani, Gavrilov, Mourtada and many others.
Due to the deficiency of fractal analysis applied directly to planar vector fields for more complicated limit periodic sets, in the second part of the thesis (Chapters 2 and 3), we consider complexified systems from the viewpoint of fractal analysis of orbits. By complexifying germs of planar vector fields at both elliptic (weak focus) and hyperbolic (saddle) singular points, we obtain germs of complex saddle vector fields in , see [26, Chapters 4 and 22]. It was noticed in  or  that the box dimension of an orbit of the Poincaré map or, equivalently, of a spiral trajectory around the elliptic point, distinguishes between weak and strong foci, which are classified by the order of the first non-zero term in their normal forms. However, planar fractal analysis fails in distinguishing between weak and strong resonant saddle points, since they are not monodromic points: there is no recurring spiral trajectory accumulating at them and the Poincaré map is not defined. In this case, we complexify the resonant saddles. The leaves of resonant complex saddles are monodromic and an analogue of the first return map, called the holonomy map, is well defined. In this case, we expect the box dimension of leaves, or of orbits of their holonomy maps, to distinguish between weak and strong saddles, which difer by the order of the first non-zero resonant term in their formal normal forms. Already Il’yashenko, in his proof of the Dulac problem about nonaccumulation of limit periodic sets on elementary planar polycycles, considered complexified systems in . Therefore, we hope that the analysis of complexified dynamics may give some insight into unsolved planar cases in the future.
An important way of classifying and recognizing germs of complex saddle vector fields are their orbital formal and analytic normal forms. We are here concerned only with orbital formal classification of complex saddles, which can be found for example in the book of Il’yashenko and Yakovenko [26, Section 22] or in the book of Loray [32, Chapter 5]. The germs of vector fields with a complex saddle are either formally orbitally linearizable or their formal normal form is described by two parameters called formal invariants. Our first goal is to see if we can read formal invariants of a complex saddle by fractal analysis of one leaf of a foliation near the origin, or, equivalently, by fractal analysis of one orbit of its holonomy map defined on a cross-section to the saddle. In the complex case, one leaf of a foliation can be understood as one trajectory of the system (in complex time), while complex holonomy map is the complex equivalent of the Poincaré map.
As was the case with Poincaré maps of planar vector fields, in complex saddle cases the analysis of holonomy maps is sufficient for classifying complex saddles. It was stated by Mattei, Moussu  that formal (analytic) orbital normal forms of germs of complex saddles can be read from formal (analytic) classes of their holonomy maps, see [32, Théorème 5.2.1]. Furthermore, by Lemma 22.2 in , holonomy maps of complex saddles are germs of complex diffeomorphisms fixing the origin, . Their formal classification was given by Birkhoff, Kimura, Szekeres and Ecalle in the mid century and can be found in [26, Section 22B] or [32, Section 1.3]. We consider all germs of diffeomorphisms except the most complicated cases of irrational rotations in the linear part. Sections 2.2 and 2.3 of this thesis are thus dedicated to establishing a bijective correspondence between the formal classification of germs of diffeomorphisms of the complex plane and the fractal properties of only one discrete orbit. The results from this chapter are mostly published in 2013 in paper  by Resman. Since the formal invariants are complex numbers, we had to generalize fractal properties to become complex numbers, revealing not only the density, but also the direction of the orbit. We call them the directed fractal properties. By definition, they are related to the directed area or the complex measure of the -neighborhood of the orbit defined here. It incorporates not only the area, but also the center of the mass of the -neighborhood.
The results are then directly applied to germs of complex saddle fields in Chapter 2.4. We read the orbital formal normal form of a saddle field, using fractal properties of its holonomy map. Furthermore, we compute the box dimension of a trajectory around a planar saddle loop, and thus give the preliminary steps for computing the box dimension of a leaf of a foliation for germs of resonant complex saddle fields. We state the conjecture connecting the dimension of one leaf of a foliation and the first formal invariant of a resonant nonlinearizable complex saddle. For linearizable resonant saddles, box dimension of a leaf should be trivial, that is, equal to 2. The conjecture has yet to be proven.
The formal classification problem for complex germs of diffeomorphisms being fully solved, in Chapter 3 we investigate how far methods of fractal analyis can bring us in the problem of analytic classification. We consider germs of parabolic diffeomorphisms. The analytic classification problem for parabolic diffeomorphism was solved by Ecalle  and Voronin  around the year 1980. From then on, many authors have been working on understanding ideas and tools from  and treating the problem from different viewpoints, for example Loray, Sauzin, Dudko etc. For a good overview, we recommend the preprint of Sauzin , the book of Loray  or recent thesis of Dudko . Most of the authors restrict to the simplest, model formal class of diffeomorphisms. We also follow this fashion. The complexity of the problem lies in the fact that the analytic class of a parabolic diffeomorphism is given by finitely many pairs of germs of diffeomorphisms in , the so-called Ecalle-Voronin moduli of analytic classification. The same can be expressed in terms of infinite sequences of numbers. The complexity of the space of analytic invariants is not unexpected. Indeed, it was shown by Ecalle that, for determining the analytic class, we need information on the whole diffeomorphism. No finite jet of a diffeomorphism is sufficient. More precisely, each parabolic diffeomorphism is formally equivalent to its formal normal form, but the formal change of variables converges only sectorially to analytic functions. The neighboring sectors overlap, resembling the petals of a flower. The analytic conjugacies on sectors are obtained as sectorial solutions of the Abel (trivialisation) difference equation for a diffeomorphism. The difference between them on the intersections of sectors is exponentially small. This is an ocurrence of the famous Stokes phenomenon, which can be overviewed in book . The Ecalle-Voronin moduli are obtained comparing the analytic solutions on intersections of sectors, and incorporate information on exponentially small differences. The moduli are not computable nor operable even in the simplest cases. Therefore, some authors restrict themselves to considering only the computable tangential derivative to the moduli, for example Elizarov in .
The approach to the problem of analytic classification using fractal properties of orbits in this thesis is new. It is still not clear whether it is possible to read the analytic moduli using -neighborhoods of orbits. The problem seems to be very difficult. In Chapter 3 of the thesis, we investigate the analyticity properties of the complex measure of -neighborhoods of orbits, as function of parameter and of initial point . We show the lack of analyticity in each variable. Nevertheless, we note that the first coefficient dependent on the initial point in the asymptotic development in of the complex measure of the -neighborhood, regarded as function of , has sectorial analyticity property. We call this coefficient the principal part of the complex measure. It satisfies the difference equation similar to the Abel (trivialisation) equation, which we call the -Abel equation. We generalize both equations introducing the generalized Abel equations, and give the necessary and sufficient conditions on a diffeomorphisms for the global analyticity of solutions of their generalized Abel equations. We apply the results to obtain examples which show that the global analyticity of principal parts is not in correlation with the analytic conjugacy of the diffeomorphism to the model. To support this statement theoretically, in a similar way as analytic classes were defined comparing sectorial solutions of Abel equation, we define a new classification of diffeomorphisms comparing the sectorial solutions of -Abel equation. Thus we obtain equivalence classes which we call the -conjugacy classes. We show that these new classes are ’far’ from the analytic classes. In fact, they are in transversal position with respect to analytic classes. This means, inside each analytic class we can find diffeomorphisms belonging to any -class. We also define higher conjugacy classes, with respect to generalized Abel equations with right-hand sides of higher orders.
0.2 The thesis overview
Here we repeat shortly by chapters the main results presented in the thesis.
Chapter 1 is dedicated to two main results published in Mardešić, Resman, Županović . They concern fractal analysis of discrete systems generated by local diffeomorphisms of the real line at a fixed point. In case of generators sufficiently differentiable at fixed point, the bijective correspondence between the multiplicity of the fixed point and the box dimension of any orbit is given in Theorem 1.1. In case of generators differentiable except at fixed point and belonging to a Chebyshev scale, we show that the box dimension of orbits cannot recognize the multiplicity precisely. Therefore we introduce the critical Minkowski order, as a generalization of box dimension, which is adapted to a given scale. In Theorem 1.3, the bijective correspondence is given between the multiplicity of a generator in a given scale and the critical Minkowski order of one orbit. At the end of Chapter 1, in Section 1.4, the results are applied to Poincaré maps for elliptic points, limit cycles and homoclinic loops. The application is in reading the cyclicity of these sets in generic unfoldings from the box dimension or the critical Minkowski order of only one orbit of their Poincaré maps.
Chapter 2 treats complex germs of diffeomorphisms , whose linear part is not an irrational rotation, and germs of resonant complex saddles in .
In Sections 2.2 and 2.3, fractal analysis of orbits is brought to relation with existing formal classification results. In Section 2.2, box dimension of orbits of hyperbolic germs is computed in Proposition 2.2 to be equal to . Its triviality is consistent with analytic linearizability of such germs. In Subsection 2.3, formal classification of parabolic diffeomorphisms is treated. The results were published in . The area of -neighborhood of orbit is generalized as the directed area or the complex measure of the -neighborhood, incorporating the area and the center of the mass of the -neighborhood. Three coefficients in its asymptotic development: box dimension, directed Minkowski content and directed residual content are introduced in a natural way. Main Theorems 2.2 and 2.3 state the bijective correspondence between the three fractal properties of any discrete orbit near the origin and the elements of the formal normal form of the generating diffeomorphism.
In Section 2.4, the results are applied to the formal orbital classification of resonant complex saddles in . In Subsection 2.4.2, the direct application of the previous results to vector fields, using their holonomy maps, in given in Proposition 2.14. In Subsection 2.4.3, in Theorem 2.4, we compute the box dimension of the spiral trajectory around the planar homoclinic loop. It is a preliminary result containing expected techniques for computing the box dimension of leaves of foliations given by vector fields in with complex saddle. In Subsection 2.4.4, we finally state a conjecture about the box dimension of a leaf of a foliation for a resonant formally nonlinearizable saddle: it is in a bijective correspondence with the first element of the formal normal form. For formally linearizable resonant saddles, we conjecture that the box dimension is 2.
In Chapter 3, we consider analytic classification of parabolic diffeomorphisms, from the viewpoint of -neighborhoods of their orbits. In Section 3.1, we make a rather long introduction about analytic classification problem from the literature, with definitions and techniques we will need. In Section 3.2, we show that the function of complex measure of -neighborhoods of orbits does not posses the analyticity property in any variable. We define the principal part of the complex measure as the first coefficient dependent on the initial point in the development in of the complex measure of the -neighborhoods of orbits. It is regarded as a function of . Theorem 3.6 states sectorial analyticity properties of this principal part and a difference equation it satisfies. We call such equations the generalized Abel equations. In Section 3.3, we consider analyticity properties of solutions of generalized Abel equations and state in Theorem 3.5 the necessary and sufficient conditions for the existence of a globally analytic solution. Finally, in Theorem 3.7, we characterize the diffeomorphisms whose principal parts of orbits are globally analytic. In Section 3.4, we compare analyticity results concerning principal parts with analytic classification results. We first show some examples that suggest that the analytic classes cannot be read from the principal parts of orbits. Then, to confirm the anticipated, we introduce a new classification of diffeomorphisms using the equation for the principal parts, called the -classification. We show finally, in Theorem 3.8 and Proposition 3.8, that the newly defined classes are in transversal position with respect to the analytic classes, meaning that inside each analytic class there exist diffeomorphisms belonging to any -class.
0.3 Main definitions and notations
Here we state main definitions and notations used throughout the thesis. The definitions and notations specific for each chapter, on the other hand, are introduced at the beginning of each chapter.
Let be a bounded set. By , , we denote its -neighborhood:
Let , , be Lebesgue measurable, and let denote its Lebesgue measure. In the thesis, the Lebesgue measure in , the length, will be denoted by , while the Lebesgue measure in or , the area, will be denoted by . The fractal properties of set are related to the asymptotic behavior of the Lebesgue measure of its -neighborhood , as . The rate of decrease of , as , reveals the density of accumulation of the set in the ambient space. It is measured by the box dimension and the Minkowski content of .
By lower and upper -dimensional Minkowski content of , , we mean
respectively. Furthermore, lower and upper box dimension of are defined by
As functions of , and are step functions that jump only once from to zero as grows, and upper or lower box dimension are equal to the value of when jump in upper or lower content appears, see Figure 1.
If , then we put and call it the box dimension of . In literature, the upper box dimension of is also referred to as the limit capacity of , see for example .
If and , we say that is Minkowski nondegenerate. If moreover , we say that is Minkowski measurable. The notion was introduced by Stachó  in 1976. In that case, we denote the common value of the Minkowski contents simply by , and call it the Minkowski content of .
In this thesis we deal only with nice sets, for which the upper and the lower box dimension and also the upper and the lower Minkowski contents coincide. Therefore, from now on, we speak only about the box dimension and the Minkowski content .
In the next example, we show how box dimension and, additionally, Minkowski content distinguish between the rates of growth of -neighborhoods and thus between densities of sets in the ambient space.
Example 0.1 (Box dimension and asymptotic behavior of -neighborhoods of sets).
If , as , , , in the sense that , then and .
If , as , , then , but , signaling that the set fills more space than in the above example.
Similarly, if , as , , then , but , signaling lower density of accumulation.
If , as , in the sense that there exist such that , then , but the upper and the lower Minkowski content do not necessarily coincide.
The sets that we study have an accumulation set. For example, the sets consist of points accumulating at the origin, of spiral trajectories accumulating at singular points or polycycles, of hyperbolas accumulating at saddles etc. Box dimension and Minkowski contents of such sets measure the density of the accumulation. In these cases, for computing the behavior of the Lebesgue measure of their -neighborhoods, we always use the direct, simple procedure that was described in book of Tricot . We divide the -neighborhood into tail and nucleus , different in geometry, and compute their behavior separately. The tail denotes the disjoint finitely many first parts of the -neighborhood, while the remaining connected part is called the nucleus.
We state some important properties of box dimension that we will use in this work, from Falconer . Let , such that exist.
(box dimension under lipschitz mappings) Let be a lipschitz mapping111There exists a constant such that . Then
In particular, if is bilipschitz222There exists constants such that , then
In particular, any diffeomorphism is a bilipschitz mapping.
(the finite stability property) . On the contrary, the countable stability property does not hold.
(monotonicity) Let . Then, .
(box dimension of the closure) .
(Cartesian product) .
Furthermore, the lipschitz property and the monotonicity property hold for the lower and the upper box dimension. The finite stability property holds only for the upper box dimension.
In the end, let us mention that the box dimension and the Minkowski content, as shown in Example 0.1, address only the first term in the asymptotic development of the Lebesque measure of the -neighborhood of the set. Further development does not matter. In this thesis, we sometimes need finer information. We sometimes refer to the whole function of the Lebesgue measure of the -neighborhoods, , as a fractal property of the set, or to its asymptotic development in up to a finite number of terms.
Let be a sequence of functions , ordered by increasing flatness at :
For example, the scale can be the power scale, , the logarithmic scale , the exponential scale , or any other scale satisfying (0.1).
The series of functions ,
without addressing the question of convergence of the series, is called the formal series. Furthermore, we say that a function has formal asymptotic development (0.2) or a formal asymptotic development in the scale , as , if, for every , it holds that
Note that the series (0.2) may or may not converge in a neighborhood of . We do not raise the question of convergence. The function develops in a given scale, but it may not be true that the series actually converges on any small neighborhood of .
In the same way, we define the asymptotic development at .
We use formal asymptotic developments and formal series many times in the thesis: asymptotic developments in Chebyshev scales in Chapter 1, formal asymptotic development of -neighborhoods, as , in Chapter 2, formal changes of variables for parabolic diffeomorphisms in Chapter 2, etc.
In complex plane , we sometimes consider a formal Taylor series at :
without addressing the question of its convergence. Usually, it is used in the context of germs333The notion of the germ refers to a function defined on some small neighborhood of the origin, not addressing the size of its domain. of formal diffeomorphisms fixing 0 or formal changes of variables, with and in (0.3). They represent a composition of countably many changes of variables of the type or , The composition may not converge.
The set of all formal series at will be denoted by . The set denotes all formal series with initial term of order or higher, .
On the other hand, if the series (0.3) converges around the origin, we call it an analytic germ. The set of all analytic germs is denoted by .
We adapt the usual convention and denote formal series by hat sign, , while convergent series are denoted simply by .
By , we denote the -jet, , of a formal series from (0.3).
Similarly as in real case, we say that a germ has formal development , as , on some open sector centered at the origin if, for every and every closed subsector , there exists a constant , such that it holds
Finally, for two real, positive germs of real variable , we write
if , for some . We write
if there exist , and , such that , for all . The same notation is used for germs at infinity.
For real or complex germs and , we write
if it holds that . We write
if there exists a constant and a punctured neighborhood of , such that it holds .
Chapter 1 Application of fractal analyis in reading multiplicity of fixed points for diffeomorphisms on the real line
1.1 Introduction and definitions
In this chapter, we consider one-dimensional discrete systems on the real line, generated by diffeomorphisms around their fixed points.
Let , , be a function with fixed point , which is a diffeomorphism on , but not necessarily at the fixed point. This function is called a generator of a dynamical system. If is (sufficiently) differentiable at the fixed point , we refer to it as case of differentiable generator. If not, we call it non-differentiable generator case.
Let . Suppose that the sequence of iterates , , remains in . This sequence is called an orbit generated by diffeomorphism , with initial point , and denoted
Changing the initial point we get a one-dimensional discrete dynamical system generated by . In this work, we consider generators whose orbits around the fixed point accumulate at the fixed point.
Fractal properties of orbit , namely its box dimension and Minkowski content, are, by definition in Chapter ‣ Chapter 0 Introduction, closely related to the asymptotic behavior of the length of the -neighborhood of the orbit, denoted , as . In this chapter, we study the relationship between the multiplicity of a fixed point of a function , and the dependence on of the length of -neighborhoods of any orbit of near the fixed point.
In Section 1.2, we consider the case of a differentiable generator. The results were mostly given by Elezović, Županović, Žubrinić in . In Section 1.3, we generalize these results to non-differentiable cases. Finally, in Section 1.4, we apply the results to Poincaré maps around monodromic limit periodic sets and to Abelian integrals. The fractal method that considers fractal properties of only one orbit of the Poincaré map is a new method in estimating cyclicity of limit periodic sets. All results of this chapter are published in Mardešić, Resman, Županović .
We recall here the basic definitions we will use in this chapter. They are mainly taken from the book of P. Mardešić about Chebyshev systems, .
Recall the standard definition of multiplicity of a fixed point of a function differentiable at the fixed point. Let , , denote the family of -diffeomorphisms on ( included), . Let have a fixed point at .
Let on . Any fixed point of becomes a zero point of .
Definition 1.1 (Multiplicity of a fixed point of a differentiable function).
We say that is a fixed point of of multiplicity , , , and denote , if it holds that
That is, if is a zero point of of multiplicity in the standard sense.
Equivalently, since , , condition (1.1) can be expressed in terms of Taylor series for . It holds that if and only if is the first monomial with non-zero coefficient in Taylor expansion of at .
Note that this definition strongly depends on sufficient differentiability of at . However, we can put the definition of multiplicity of fixed point for differentiable functions in more general context of multiplicity of fixed point within some family of functions. This definition does not depend on differentiability, and can therefore be generalized to functions non-differentiable at the fixed point. In fact, multiplicity of a fixed point of denotes the number of fixed points that bifurcate from the fixed point in small bifurcations of within the differentiable family . This motivates the following definition:
Definition 1.2 (Multiplicity of a fixed point within a family of functions, Definition 1.1.1 in ).
Let . Let be a topological space of parameters and let , , be a family of functions, such that , for some . Let have an isolated fixed point at . We say that is a fixed point of multiplicity of function in the family of functions if is the largest possible integer, such that there exists a sequence of parameters , as , such that, for every , has distinct fixed points different from and , as , . We write
If such does not exist, we say that .
If we denote , , the above definition can also be expressed as multipicity of zero point of function in the family of functions .
Note that the multiplicity from Definition 1.2 depends on the family within which we consider function . If , then obviously
Example 1.1 ().
(differentiable case) Let , with fixed point . It holds that
Here, the metric space of parameters is , with the distance function .
(non-differentiable case) Let denote the family of all functions with asymptotic development111See the definition of asymptotic development in Chapter ‣ Chapter 0 Introduction., as , in the scale
where and , , extended to zero continuously by , . Let the family be derived from family the in the usual manner, i.e. .
Let , , be such that , as , for some the first monomial with nonzero coefficient in the asymptotic development of is . Then,
For example, if , then . On the other hand, if we consider in the subfamily unfolding in the subscale , we get smaller multiplicity
We dedicate a paragraph to the proof of the differentiable case The proof is important and illustrative, since it shows how fixed points bifurcate from a fixed point of multiplicity greater than zero.
Proof of case 1.([34, Example 1.1.1])
First, let function have a zero point of multiplicity bigger than or equal to in the family . By Definition 1.2 and by Rolle’s theorem applied times, passing to limit we conclude that , .
Conversely, suppose that , . The first monomial in expansion of is then or of higher order. Let , , as . We construct a sequence
where are chosen small enough that , and, moreover, that each has different zeros in .
First, we construct in steps from , adding and missing monomials one by one, with appropriately chosen coefficients. Take , where is small enough such that and . Then, take , with small enough, such that , and that has one zero point in different from zero (possible by inverse function theorem applied to ). We continue in this fashion up to , adding the last monomial . Obviously, and we constructed different zeros in . The same can be repeated for and ,
We note in this construction that, for constructing zero points bifurcating from zero point of , we need degrees of freedom ( powers up to the first monomial in , whose coefficients we then choose freely). That is, we need to consider -parameter bifurcations of (of codimension ).
Proof of is done following the same idea, but we have to introduce generalized derivatives that act on nondifferentiable scale in the same manner as standard derivatives act on power scale. We will introduce generalized derivatives below.
Differentiable generators that belong to the class , unfold by Taylor formula in the scale of powers, . In our study of non-differentiable generators, we restrict ourselves to special classes, which have the asymptotic development in Chebyshev scales. The definition of the Chebyshev scale is based on the notion of extended complete Chebyshev (or Tchebycheff) systems (ECT-s), see  and . The notion of asymptotic Chebyshev scale was introduced by Dumortier, Roussarie in .
Definition 1.3 (Chebyshev scale).
A finite or infinite sequence of functions of the class , , is called a Chebyshev scale if the following holds:
A system of differential operators , , is well defined on inductively by the following division and differentiation algorithm:
for every , except possibly at , to which they are extended by continuity.
The functions are strictly increasing on , .
, for , .
We call the -th generalized derivative of in the scale .
Example 1.2 (Examples of Chebyshev scales).
differentiable case: ,
More generally, any scale of monomials of the type , ordered by increasing flatness:
We say that a function has a development of order in Chebyshev scale if there exist coefficients , such that
and the generalized derivatives , , verify (in the limit sense). Similarly, we say that has an asymptotic development in Chebyshev scale if there exists a sequence such that for every there exists such that (1.2) holds. Note that this is just a reformulation of definition of asymptotic development in a scale from Section 0.3 for Chebyshev scales.
The generalized derivatives act on Chebyshev scales in the same way as standard derivatives act on the power scale: the -th generalized derivative annulates the first monomials of the Chebyshev scale. Therefore, in the asymptotic development (1.2) above,
Furthermore, it is equivalent
We say that a parametrized family has a uniform development of order in a family of Chebyshev scales , if there exist coefficients , , such that it holds
and the generalized derivatives , , verify , in the limit sense uniformly with respect to .
The following lemma is a generalization of the statement from Example 1.1,1., where differentiable case was considered. It is a combination of results from Lemma 1.2.2 in  and Joyal’s Theorem 21. in .
Lemma 1.1 (Generalized derivatives and multiplicity).
Let be a family of functions having a uniform development of order in a family of Chebyshev scales , . Let . Let be derived from in the usual way, . If the generalized derivatives of at satisfy
(that is, if is the first nonzero coefficient in the development of ), then
Moreover, if , , and the matrix
is of maximal rank , then (1.4) is equivalent to .
Idea of the proof. Proof is similar as in Example 1.1, using generalized derivatives instead of standard. The distance function on is given analogously by , .
One direction follows as before by Rolle’s theorem. The contrary does not hold without regularity assumption (1.5). The difficulty here is that we are restricted by the family in the choice of small deformations that, added to , need to generate small zeros. Nevertheless, if condition (1.5) is satisfied, by the implicit function theorem we get freedom in choice of small coefficients , by expressing parameters , as functions of independent variables .
1.2 Generators differentiable at a fixed point
In this section, we consider generators sufficiently differentiable at a fixed point and state a bijective correspondence between the multiplicity of the fixed point and the box dimension of any orbit tending to the fixed point. The results are just a reformulation of results from Elezović, Županović, Žubrinić (see Theorems 1, 5 and Lemma 1 in ).
Theorem 1.1 (Multiplicity of fixed points and box dimension of orbits, differentiable case, Theorem 1 from  reformulated).
Let be sufficiently differentiable on , and positive on . Let and let be any orbit with initial point sufficiently close to 0.
If , then it holds
If and additionally on , then it holds that
Moreover, for , a bijective correspondence holds
By Taylor formula applied to a sufficiently differentiable function , we get . Therefore we are under assumptions of Theorems 1 and 5 from  and the dimension result (1.8) follows from these theorems. However, the asymptotic development of was not explicitely computed there, therefore we do it here.
We estimate the length directly, dividing the -neighborhood of in two parts: the nucleus, , and the tail, . This way of computing was suggested by Tricot . The tail is the union of all disjoint -intervals of the -neighborhood, before they start to overlap. It holds that
Let denote the critical index separating the tail and the nucleus. It describes the moment when -intervals around the points start to overlap. The critical index is well-defined since the points of orbit tend to zero with strictly decreasing distances between consecutive points. We have that
Denote by the distances between consecutive points. To compute asymptotic behavior of , as , we have to solve (to first term only)
By Theorem 1 in , in case the points of orbit and their distances have the following asymptotic behavior:
In case and , it either holds
, if , or
, , if .
Directly iterating , and since , we get the following estimates
for and some positive constants and .
We illustrate further computations only in case