Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

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Dynamics of quadratic polynomials, I–II

We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon. Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.

/pdf/almost-sure-invariance-principle-for-nonuniformly-hyperbolic-53x7gao1gz.pdf

Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems

We describe a method for proving subexponential lower bounds for correlations functions, and apply it to study decay of correlations for maps with countable Markov partitions. One result is that LS Young’s upper estimates [Y2] are optimal in many situations. Our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on Banach spaces, which we apply to the iterates of the transfer operator.

Subexponential decay of correlations

Linearization in the large of nonlinear systems and Koopman operator spectrum

We consider multimodal C3 interval maps f satisfying a summability condition on the derivatives Dn along the critical orbits which implies the existence of an absolutely continuous f-invariant probability measure mu. If f is non-renormalizable, mu is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence (Dn) as n → infinity. We also give sufficient conditions for mu to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.

/pdf/decay-of-correlations-in-one-dimensional-dynamics-3490imvn2z.pdf

Decay of correlations in one-dimensional dynamics

Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of infinity. We call a C1 endomorphism of the compact interval (or the circle) hyperbolic if it has finitely many hyperbolic attracting periodic points and the complement of the basin of attraction of these points is a hyperbolic set. By a theorem of Mañé for C2 maps, this is equivalent to the following conditions: all periodic points are hyperbolic and all critical points converge to periodic attractors. Note that the space of hyperbolic maps is an open subset in the space of real polynomials of fixed degree, and that every hyperbolic map satisfying the mild “no-cycle” condition (which states that orbits of critical points are disjoint) is structurally stable; see [dMvS93]. Theorem 1 solves the 2nd part of Smale’s eleventh problem for the 21st century [Sma00]:

Density of hyperbolicity in dimension one

On considere la classe #7B-A des applications C ∞ f:[0,1]→[0,1] telles que f(0)=f(1)=0 et f a un unique point critique C∈(0,1). Si le point critique de f∈#7B-A est non plat alors f n'a pas d'intervalle errant

https://www.jstor.org/stable/info/1971516

A structure theorem in one dimensional dynamics

T I lHE scientific community dealing with earth characteristics, marine and air navigation, and modern concepts of national defense is vitally concerned with longitudinal ties between continents. Expanded interest in modern geodesy has resulted particularly from research programs in developing intercontinental guided missiles and earth-circling satellites. The art of navigation has always been the most practical application of longitude, but in this the art does not require any high degree of precision. Concomitant to the age of exploration and rapid expansion of water-borne traffic was the urgent demand for a means of determining longitude at sea. Beginning with Harrison's chronometer, steady refinements were made until modern times, when continental ties are possible with the aid of telegraph and radio. Aside from navigational considerations, several purely scientific requirements involve the precise determination of longitude with reference to a standard meridian, such as Greenwich. The astronomer requires an accurate position for the time coordination of world-wide astronomical observations; the geophysicist needs longitudes of the highest precision in his studies of the drift of continents; and, finally, the geodesist desires to place all triangulation datums as closely as possible in their proper relation one with the other. Longitudinal studies, no matter how localized in place or how distant in time, have an important bearing on the subject. In celebrating the sesquicentennial of the United States Coast and Geodetic Survey during 1957, it is fitting to recall the Survey's contribution to the establishment of precise longitude in the United States. Early in the history of this bureau the need became apparent for a precise tie between some main-scheme triangulation stations in the United States and the meridians of any and all of the European observatories. The Harvard Observatory at Cambridge was adopted for the point of origin in the United States and in due time was tied to Greenwich by the method discussed in this paper. The second superintendent of the Coast Survey, Alexander Dallas Bache, was well aware of the difficulties other nations had encountered in fixing the

Wild Cantor attractors exist

We study the expansion of derivatives along orbits of real and complex one-dimensional mapsf, whose Julia setJ
 f attracts a finite setCrit of non-flat critical points. Assuming that for eachceCrit, either |D f
 n(f(c))|→∞ (iff is real) orb
 n·|Df
 n(f(c))|→∞ for some summable sequence {b
 n} (iff is complex; this is equivalent to summability of |D f
 n(f(c))|−1), we show that for everyxeJ
 f\U
 i
 f
 −i(Crit), there existl(x)≤max
 c
 l(c) andK′(x)>0
 
$$|Df^n (x)^{l(x)} \ge K^1 (x).\prod\limits_{i = 0}^{s - 1} {(K_i .|} Df^{n_i - n_i + 1} (f(c_i ))|)$$

 for infinitely manyn. Here 0=n
 s<…<n
 1<n
 0=n are so-called critical times,c
 i is a point inCrit (or a repelling periodic point in the boundary of the immediate basin of a hyperbolic periodic attractor), which shadows orb(x) forn
 i−ni
 +1 iterates, and
 
$$D_k (ci) = \left\{ {\begin{array}{*{20}c} {\max (\lambda ,K.|Df^k (f(c_i ))|)} \\ {\max (\lambda ,K.b_k .|Df^k (f(c_i ))|)} \\\end{array}} \right\}\begin{array}{*{20}c} {if f is real,} \\ {if f is complex,} \\\end{array}$$

 , for uniform constantsK>0 and λ>1. If allceCrit have the same critical order, thenK′(x) is uniformly bounded away from 0. Several corollaries are derived. In the complex case, eitherJ
 f=
 
$$\hat C$$

 orJ
 f has zero Lebesgue measure. Also (assuming all critical points have the same order) there existk>0 such that ifn is the smallest integer such thatx enters a certain critical neighbourhood, then |Df
 n(x)|≥k.

S. van Strien

Papers

Decay of correlations in one-dimensional dynamics

Density of hyperbolicity in dimension one

A structure theorem in one dimensional dynamics

Wild Cantor attractors exist

Expansion of derivatives in one-dimensional dynamics