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Discrete & Continuous Dynamical Systems - A

2006 , Volume 15 , Issue 1

Special Issue
Ergodic Theory and Non-uniform Dynamical Systems

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Xavier Bressaud, Y. Lacroix, Carlangelo Liverani and Sandro Vaienti
2006, 15(1): i-ii doi: 10.3934/dcds.2006.15.1i +[Abstract](55) +[PDF](33.0KB)
As the field of ergodic theory has branched out in considerably many directions it has become harder, even for the expert, to keep a unitary vision of the field.
   From May 17th, 2004 to May 28th, 2004 the following were held in Marseille, CIRM; first a one week school on ergodic theory devoted to the presentation of several topics of the field, second, a one week conference focused on the recent advances in the study of non-uniformly hyperbolic dynamical systems, in connection with smooth ergodic theory.
   The idea then emerged of the edition of a special issue collecting contributions to this two week event, providing a vast panorama on ergodic theory especially addressed to people working in concrete smooth dynamical systems.

For more information please click the “Full Text” above.
A survey of recent results on some statistical features of non-uniformly expanding maps
José F. Alves
2006, 15(1): 1-20 doi: 10.3934/dcds.2006.15.1 +[Abstract](110) +[PDF](283.7KB)
The aim of this paper is to present a survey of recent results on the statistical properties of non-uniformly expanding maps on finite-dimensional Riemannian manifolds. We will mostly focus on the existence of SRB measures, continuity of the SRB measure and its entropy, decay of correlations, and stochastic stability.
Smoothness of solenoidal attractors
Artur Avila, Sébastien Gouëzel and Masato Tsujii
2006, 15(1): 21-35 doi: 10.3934/dcds.2006.15.21 +[Abstract](205) +[PDF](272.0KB)
We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$:

$\ T:S^{1}\times \R\to S^{1}\times \R,\qquad T(x,y)=(l x, \lambda y+f(x)) \

where l ≥ 2, $0<\lambda<1$ and $f$ is a $C^{r}$ function on $S^{1}$. We show that, if $\lambda^{1+2s}l>1$ for some $0\leq s< r-2$, the density of the SBR measure for $T$ is contained in the Sobolev space $W^{s}(S^{1}\times \R)$ for almost all ($C^r$generic, at least) $f$.

Hyperbolicity in multi-dimensional Hamiltonian systems with applications to soft billiards
Péter Bálint and Imre Péter Tóth
2006, 15(1): 37-59 doi: 10.3934/dcds.2006.15.37 +[Abstract](99) +[PDF](277.7KB)
When considering hyperbolicity in multi-dimensional Hamiltonian sytems, especially in higher dimensional billiards, the literature usually distinguishes between dispersing and defocusing mechanisms. In this paper we give a unified treatment of these two phenomena, which also covers the important case when the two mechanisms mix. Two theorems on the hyperbolicity (i.e. non-vanishing of the Lyapunov exponents) are proven that are hoped to be applicable to a variety of situations.
   As an application we investigate soft billiards, that is, replace the hard core collision in dispersing billiards with disjoint spherical scatterers by motion in some spherically symmetric potential. Analogous systems in two dimensions have been widely investigated in the literature, however, we are not aware of any mathematical result in this multi-dimensional case. Hyperbolicity is proven under suitable conditions on the potential. This way we give a natural generalization of the hyperbolicity results obtained before in two dimensions for a large class of potentials.
Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies
Yongluo Cao, Stefano Luzzatto and Isabel Rios
2006, 15(1): 61-71 doi: 10.3934/dcds.2006.15.61 +[Abstract](127) +[PDF](247.4KB)
We study the hyperbolicity of a class of horseshoes exhibiting an internal tangency, i.e. a point of homoclinic tangency accumulated by periodic points. In particular these systems are strictly not uniformly hyperbolic. However we show that all the Lyapunov exponents of all invariant measures are uniformly bounded away from 0. This is the first known example of this kind.
k-limit laws of return and hitting times
V. Chaumoître and M. Kupsa
2006, 15(1): 73-86 doi: 10.3934/dcds.2006.15.73 +[Abstract](87) +[PDF](259.5KB)
We characterize the set Ḟ of possible $k$-limit laws of return times which appears to be independent of $k$. We construct a rank-one system having all the functions of Ḟ as a $k$-limit law of return times. We exhibit a link between $k$-limit laws of return and hitting times. We conclude with a discussion over the $n$-uples ($1$-limit law, ..., $n$-limit law) of return times.
Ergodic properties of signed binary expansions
Karma Dajani, Cor Kraaikamp and Pierre Liardet
2006, 15(1): 87-119 doi: 10.3934/dcds.2006.15.87 +[Abstract](113) +[PDF](480.5KB)
In this paper it is shown that the classical signed binary expansion involves mainly two dynamical systems: the binary odometer and a three state Markov chain. Introducing the notions of additive and multiplicative block functions (e.g., sum-of-digits and Hamming weight functions), we derive dynamical systems which are skew products over the odometer. Their spectral properties are investigated, and applications are given to certain Maharam extensions. The proofs are related to the spectral measure of unitary operators, obtained from cocycles associated to block functions.
An introduction to joinings in ergodic theory
Thierry de la Rue
2006, 15(1): 121-142 doi: 10.3934/dcds.2006.15.121 +[Abstract](106) +[PDF](334.5KB)
Since their introduction by Furstenberg [3], joinings have proved a very powerful tool in ergodic theory. We present here some aspects of the use of joinings in the study of measurable dynamical systems, emphasizing
  • the links between the existence of a non trivial common factor and the existence of a joining which is not the product measure,
  • how joinings can be employed to provide elegant proofs of classical results,
  • how joinings are involved in important questions of ergodic theory, such as pointwise convergence or Rohlin's multiple mixing problem.
Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''
Yves Derriennic
2006, 15(1): 143-158 doi: 10.3934/dcds.2006.15.143 +[Abstract](130) +[PDF](265.0KB)
This paper gives a redaction of a talk delivered at the "Ecole pluri-thématique de théorie ergodique '' which took place at the CIRM of Marseille in May 2004.
Decay of correlations for non-Hölder observables for one-dimensional expanding Lorenz-like maps
Karla Díaz-Ordaz
2006, 15(1): 159-176 doi: 10.3934/dcds.2006.15.159 +[Abstract](104) +[PDF](277.2KB)
We prove that a one-dimensional expanding Lorenz-like map admits an induced Markov structure which allows us to obtain estimates for the rates of mixing for observables with weaker regularity than Hölder.
C *-algebras associated to dynamical systems
Søren Eilers
2006, 15(1): 177-192 doi: 10.3934/dcds.2006.15.177 +[Abstract](103) +[PDF](327.1KB)
We give an overview of important areas of contact between dynamical systems and operator algebras in the context of classification, describing two different invariant ways of associating a C *-algebra to certain dynamical systems and comparing them in the case of substitutional shift spaces.
Cohomology free systems and the first Betti number
Federico Rodriguez Hertz and Jana Rodriguez Hertz
2006, 15(1): 193-196 doi: 10.3934/dcds.2006.15.193 +[Abstract](101) +[PDF](171.6KB)
We prove that a cohomology free flow on a manifold $M$ fibers over a diophantine translation on $\T^{\beta_1}$ where $\beta_1$ is the first Betti number of $M$.
Ergodic Optimization
Oliver Jenkinson
2006, 15(1): 197-224 doi: 10.3934/dcds.2006.15.197 +[Abstract](195) +[PDF](417.2KB)
Let $f$ be a real-valued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic $f$-average is as large as possible.
   In these notes we establish some basic aspects of the theory: equivalent definitions of the maximum ergodic average, existence and generic uniqueness of maximizing measures, and the fact that every ergodic measure is the unique maximizing measure for some continuous function. Generic properties of the support of maximizing measures are described in the case where the dynamics is hyperbolic. A number of problems are formulated.
Existence and uniqueness of maximizing measures for robust classes of local diffeomorphisms
Krerley Oliveira and Marcelo Viana
2006, 15(1): 225-236 doi: 10.3934/dcds.2006.15.225 +[Abstract](132) +[PDF](229.4KB)
We prove existence of maximal entropy measures for an open set of non-uniformly expanding local diffeomorphisms on a compact Riemannian manifold. In this context the topological entropy coincides with the logarithm of the degree, and these maximizing measures are eigenmeasures of the transfer operator. When the map is topologically mixing, the maximizing measure is unique and positive on every open set.
Dynamical properties of logical substitutions
Giovanni Panti
2006, 15(1): 237-258 doi: 10.3934/dcds.2006.15.237 +[Abstract](123) +[PDF](979.5KB)
Many kinds of algebraic structures have associated dual topological spaces, among others commutative rings with $1$ (this being the paradigmatic example), various kinds of lattices, boolean algebras, C *-algebras, .... These associations are functorial, and hence algebraic endomorphisms of the structures give rise to continuous selfmappings of the dual spaces, which can enjoy various dynamical properties; one then asks about the algebraic counterparts of these properties. We address this question from the point of view of algebraic logic. The datum of a set of truth-values and a "conjunction'' connective on them determines a propositional logic and an equational class of algebras. The algebras in the class have dual spaces, and the duals of endomorphisms of free algebras provide dynamical models for Frege deductions in the corresponding logic.
Recurrence rate in rapidly mixing dynamical systems
Benoît Saussol
2006, 15(1): 259-267 doi: 10.3934/dcds.2006.15.259 +[Abstract](157) +[PDF](246.6KB)
For measure preserving dynamical systems on metric spaces we study the time needed by a typical orbit to return back close to its starting point. We prove that when the decay of correlation is super-polynomial the recurrence rates and the pointwise dimensions are equal. This gives a broad class of systems for which the recurrence rate equals the Hausdorff dimension of the invariant measure.
A notion of independence via moving targets
Jörg Schmeling
2006, 15(1): 269-280 doi: 10.3934/dcds.2006.15.269 +[Abstract](105) +[PDF](229.6KB)
We introduce a new notion of independence based on the Borel--Cantelli lemma. We study this characteristic in the context of i.i.d. stochastic processes and processes driven by equilibrium dynamics.
Euclidean dynamics
Brigitte Vallée
2006, 15(1): 281-352 doi: 10.3934/dcds.2006.15.281 +[Abstract](101) +[PDF](812.8KB)
We study a general class of Euclidean algorithms which compute the greatest common divisor [gcd], and we perform probabilistic analyses of their main parameters. We view an algorithm as a dynamical system restricted to rational inputs, and combine tools imported from dynamics, such as transfer operators, with various tools of analytic combinatorics: generating functions, Dirichlet series, Tauberian theorems, Perron's formula and quasi-powers theorems. Such dynamical analyses can be used to perform the average-case analysis of algorithms, but also (dynamical) analysis in distribution.
Asymptotic orbit complexity of infinite measure preserving transformations
Roland Zweimüller
2006, 15(1): 353-366 doi: 10.3934/dcds.2006.15.353 +[Abstract](121) +[PDF](279.9KB)
We determine the asymptotics of the Kolmogorov complexity of symbolic orbits of certain infinite measure preserving transformations. Specifically, we prove that the Brudno - White individual ergodic theorem for the complexity generalizes to a ratio ergodic theorem analogous to previously established extensions of the Shannon - McMillan - Breiman theorem.

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