Geometric theory of dynamical systems

Our aim is to show that the Julia-Fatou--Sullivan structure theory for the dynamics of rational maps is also valid for smooth endomorphisms of the circle (and of the interval) under extremely mild smoothness and non-flatness conditions. In order to stress the similarity between real and complex one-dimensional dynamics let us recall the main results from the Fatou-Julia-Sullivan theory. If f is a rational map then there is a dynamical decomposition of the Riemann sphere into the disjoint union of two totally invariant (i.e., both forward and backward invariant) sets J(f), F(f). Here, F(f) is the domain of normality of the family of iterates of f , and is called the Fatou set. Its complement, which is called the Julia set off , is a compact set, which contains all the complications of the dynamics off . The connected components of the open set F(f) are mapped onto each other by f. Hence the orbit of a component of F(f) is the union of some components of F(f). Julia proved in the beginning of the century that if a component of F(f) is periodic and contains an attracting periodic point then the orbit of this component must contain a critical point. Sullivan, in the remarkable paper [Su], showed via quasi-conformal deformations that the components are eventually periodic and fall into finitely many orbits. Let N be either the circle S J or a compact interval of the real line and f." N---~N be a smooth endomorphism. A critical point of f is a point where the derivative vanishes. A critical point is non-flat if some (higher) derivative is non-zero. A critical point is an inflection point if it has a neighbourhood where f is monotone. Otherwise it is called a turning point. Assume f is not a homeomorphism ( i f f is a homeomorphism then these maps correspond to degree + 1 rational maps of the Riemann sphere and the situation is

https://projecteuclid.org/download/pdf_1/euclid.acta/1485890671

Julia-Fatou-Sullivan theory for real one-dimensional dynamics

1 Definitions and Examples: Preliminaries Basic constructions Basic examples 2 Poncare-Bendixon's theory: Existence of closed transversal Absence of non-trivial recurrent trajectories on some surfaces Hilmy's and Cherry's theorems on quasiminimal sets Gutierrez's structure theorem Limit set of individual trajectory 3 Decomposition of flows: Decomposition theorems Center of flow Blowing-down of flows Regular flows Application: smoothing of flows 4 Local Theory: Topological normal forms Analytical normal forms Smooth normal forms Finitely smooth normal forms, Degenerate critical points C1 normal forms of degenerate singularities 5 Space of Flows and vector fields: Structural stability Classification of Morse-Smale flows Lyapunov's method, Connected components of Morse-Smale flows Degrees of non-stability, Typical properties of non-stable flows 6 Ergodic theory: Liouville's theorem Kolmogorov's theorem for flows on torus Non-trivial invariant measures Ergodicity Mixing Entropy 7 Invariants of surface flows: Topological classification of torus flows Oriented surfaces of higher genus z 2 Application of geodesic laminations Transitive flows on non-orientable surfaces Classification of exceptional minimal sets Classification of regular flows Classification of non-wandering flows Cayley graph of a flow Homology and cohomology invariants Rotation sets of surface flows Smooth classification of flows 8 C*-algebras of Surface Flows: Irrational rotation algebra Artin's rotation algebra K-theory C*-algebra of Morse-Smale flows 9 Semi-local Theory: Denjoy's and Schwarz's theorems Cherry's problem Local structure preventing quasiminimality 10 Anosov-Weil Problem: Theorems of Weil and Anosov Asymptotic direction of individual curves Approximation of curve by trajectories of a flow Limit sets of curves and trajectories at the absolute Deviation of curves from the geodesics Examples of unbounded deviation 11 Non-compact Surfaces: Kaplan classification Level curves of harmonic functions Markus's classification Structural stability Neumann's example Inaba example and Beniere-Meigniez theorem, Beniere-Hector theorem Aranson-Zhuzhoma's example 12 Triptych: Geodesic frameworks revisited On continuity and collapse of geodesic frameworks Cr-Closing lemma

/pdf/flows-on-2-dimensional-manifolds-an-overview-1muvwli1m5.pdf

Flows on 2-dimensional Manifolds: An Overview

The existence of smooth families of Lorenz maps exhibiting all possible dynamical behavior is established and the structure of the parameter space of these families is described.

/pdf/universal-models-for-lorenz-maps-12ayjt08bu.pdf

Universal models for Lorenz maps

A survey of works concerning a kind of generalization of the Poincare rotation number for a rather general class of flows on closed surfaces is given. The idea (going back to A. Weil) is to replace this number by the asymptotic direction at infinity of the trajectory lifted to the universal covering plane. Other works on flows on surfaces are also mentioned, but this is done primarily for comparison in order to shed light on the subject.

Flows on closed surfaces and behavior of trajectories lifted to the universal covering plane

The purpose of this research is to describe all smooth vector fields on the torus T2 with a finite number of singularities, no periodic orbits and no saddleconnections. In this paper we are able to complete the description within the class of vector fields which are area contracting near all singularities. In particular we give a large class of analytic vector fields on the torus T2 which have non-trivial recurrence and also sinks.

On Cherry flows

A Metric property of Cherry Vector Fields on the torus

On stability of dynamical systems on open manifolds

In this note it is shown that the sum of two homogeneous Cantor sets is often a uniformly contracting self-similar set and it is given a sufficient condition for such a set to be of Lebesgue measure zero (in fact, of Hausdorff dimension less than one and positive Hausdorff measure at this dimension). 1. Definitions and results The study of the arithmetic difference (sum) of two Cantor sets has been of great interest for the homoclinic bifurcations of dynamical systems, since the last decade. For a detailed presentation of this subject we refer the reader to [7]. About 1987, J. Palis made the following Conjecture. If the sum of two affine Cantor sets has positive Lebesgue measure, then it contains an interval. Many articles have been written on this conjecture; see [4], [5], [6] and [8]. Another subject which has been of great interest since the work of B. Mandelbrot ([3]) is the study of fractal dimensions, introduced by Hausdorff in the twenties. The most useful source of examples in this context is given by self-similar sets. Several efforts have been made ([1], [2], [9]) to solve the following Question. Is it true that if a self-similar set has positive Lebesgue measure, then it contains an interval? Our first result (Theorem A below) gives an indication that the conjecture and the question above are related to each other. We are going to state it precisely. Let fi : x ∈ R 7→ cix + ai ∈ R, 0 < ci < 1, i = 1, . . . , r, be affine contractions of the real line. There is a unique compact and nonempty set A ⊂ R such that A = f1(A) ∪ · · · ∪ fr(A) ([2]). This set is called a self-similar set, and it is called a uniformly contracting self-similar set whenever c1 = · · · = cr = c. Let H(2r + 1) be the set of all vectors ~λ = (λ, λ′1, λ, . . . , λ′r, λ) ∈ R, such that λ > 0, λi > 0, i = 1, . . . , r; and λ + λ ′ 1 + λ + · · · + λr + λ = 1. Let C(~λ) be the self-similar set obtained, as above, using the contractions fi(x) = λx + ai, with a0 = 0, ai = λ+λ1 +λ+ · · ·+λi, i = 1, . . . , r. The set C(~λ) is called a homogeneous Cantor set. We have the following Received by the editors February 6, 1998. 1991 Mathematics Subject Classification. Primary 28A78, 58F14.

/pdf/sum-of-cantor-sets-self-similarity-and-measure-n5l63plc8v.pdf

Sum of Cantor sets: Self-similarity and measure

We consider the class of diffeomoIphisms, on compact two-dimensional manifolds, which are invanant under the action of a compact Lie group C and whose nonwandering set consists of a finite number of G-orbits. We descnbe the modulus of stability of almost all diffeomorphisms in this class. Introduction and statement of results. Let G be a compact Lie group acting smoothly on a smooth closed connected two-dimensional manifold M. We say that a G-equivariant diffeomorphism of M has simple recurrences if its nonwandering set consists of a finite number of normally hyperbolic critical elements. Denote bySthe set of G-equivariant diffeomorphisms with simple recurrences whose stable and unstable manifolds of its critical elements are G-transversal. There are many diffeomorphisms in Swhich are not structurally stable. However, for some of them it may be possible to parametrize all the topological conjugacy classes in their neighborhood by finitely many real parameters. In this case we say that they have finite modulus of stability. Otherwise, their modulus of stability is infinite. In this paper we study the modulus of stability of diffeomorphisms in a dense set cS. Now we will state our result in a precise way. A C-diffeomorphismf: M M is G-equivariant if f(g x) = g f(x) for all x E M and g E G. The set of such diffeomorphisms with the CX topology is denoted by DiffG(M). An element t in DiffG(M) induces a G x Z-action on M given by (g, n) * x = g * tn(x). A critical element of t is a compact G x Z-orbit. A critical element a of f is normally hyperbolic if there is a continuous invariant splitting of the tangent bundle of M over a, TaM = To @ Es (E3 EU, and constants c > 0, 0 O. The stable manifold of a, Ws(a) = {y E M: distance offn(y) to a tends to zero when n > + oo }, has G, t-invariant foliation gSS( f ) whose leaves are the strong stable manifolds of the points in a. Similarly, the unstable manifolds of a have a G, f-invariant foliation SaUu(f). Here, in dimension two, we have that a is either a periodic orbit or a finite union of circles. In the first case a is just a hyperbolic orbit and SoSS(f ) (Eau"(f )) is a finite number of submanifolds. In the second case a is an Received by the editors April 13, 1984 and, in revised form, August 21, 1984. 1980 Muthematics S"bject Clczssification. Primary 58F10; Secondary 57S15. Kez words clnd phrclses. Compact Lie groups, equivariant diffeomorphisms, structural stability, modulus of stability. l The second and third authors were partially supported by CNPq (Brazil). C)1985 American Mathematical Society 0002-9947/85 $1.00 + $ 25 per page

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Pedro Mendes

Papers

On Cherry flows

A Metric property of Cherry Vector Fields on the torus

On stability of dynamical systems on open manifolds

Sum of Cantor sets: Self-similarity and measure

Equivariant diffeomorphisms with simple recurrences on two-manifolds