ISSN:

1078-0947

eISSN:

1553-5231

All Issues

## Discrete & Continuous Dynamical Systems - A

2006 , Volume 15 , Issue 1

Special Issue

Ergodic Theory and Non-uniform Dynamical Systems

**
2006, 15(1): i-ii
doi: 10.3934/dcds.2006.15.1i
**

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.
2006, 15(1): 1-20
doi: 10.3934/dcds.2006.15.1

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.
2006, 15(1): 21-35
doi: 10.3934/dcds.2006.15.21

We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$:
2006, 15(1): 37-59
doi: 10.3934/dcds.2006.15.37

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.
2006, 15(1): 61-71
doi: 10.3934/dcds.2006.15.61

We study the hyperbolicity of a class of horseshoes exhibiting an
2006, 15(1): 73-86
doi: 10.3934/dcds.2006.15.73

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.
2006, 15(1): 87-119
doi: 10.3934/dcds.2006.15.87

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.
2006, 15(1): 121-142
doi: 10.3934/dcds.2006.15.121

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
2006, 15(1): 143-158
doi: 10.3934/dcds.2006.15.143

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.
2006, 15(1): 159-176
doi: 10.3934/dcds.2006.15.159

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.
2006, 15(1): 177-192
doi: 10.3934/dcds.2006.15.177

We give an overview of important areas of contact between dynamical systems and operator algebras in the context of
2006, 15(1): 193-196
doi: 10.3934/dcds.2006.15.193

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 $M$.
2006, 15(1): 197-224
doi: 10.3934/dcds.2006.15.197

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.
2006, 15(1): 225-236
doi: 10.3934/dcds.2006.15.225

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.
2006, 15(1): 237-258
doi: 10.3934/dcds.2006.15.237

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.
2006, 15(1): 259-267
doi: 10.3934/dcds.2006.15.259

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.
2006, 15(1): 269-280
doi: 10.3934/dcds.2006.15.269

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.
2006, 15(1): 281-352
doi: 10.3934/dcds.2006.15.281

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.
2006, 15(1): 353-366
doi: 10.3934/dcds.2006.15.353

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.

Select all articles

Export/Reference:

*+*[Abstract](35)

*+*[PDF](33.0KB)

**Abstract:**

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.

*+*[Abstract](73)

*+*[PDF](283.7KB)

**Abstract:**

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.

*+*[Abstract](142)

*+*[PDF](272.0KB)

**Abstract:**

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) $f$.

*+*[Abstract](65)

*+*[PDF](277.7KB)

**Abstract:**

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.

*+*[Abstract](92)

*+*[PDF](247.4KB)

**Abstract:**

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.

*+*[Abstract](66)

*+*[PDF](259.5KB)

**Abstract:**

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.

*+*[Abstract](74)

*+*[PDF](480.5KB)

**Abstract:**

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.

*+*[Abstract](72)

*+*[PDF](334.5KB)

**Abstract:**

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.

*+*[Abstract](87)

*+*[PDF](265.0KB)

**Abstract:**

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.

*+*[Abstract](69)

*+*[PDF](277.2KB)

**Abstract:**

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.

*+*[Abstract](70)

*+*[PDF](327.1KB)

**Abstract:**

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.

*+*[Abstract](65)

*+*[PDF](171.6KB)

**Abstract:**

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 $M$.

*+*[Abstract](146)

*+*[PDF](417.2KB)

**Abstract:**

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.

*+*[Abstract](82)

*+*[PDF](229.4KB)

**Abstract:**

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.

*+*[Abstract](80)

*+*[PDF](979.5KB)

**Abstract:**

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.

*+*[Abstract](107)

*+*[PDF](246.6KB)

**Abstract:**

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.

*+*[Abstract](72)

*+*[PDF](229.6KB)

**Abstract:**

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.

*+*[Abstract](76)

*+*[PDF](812.8KB)

**Abstract:**

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.

*+*[Abstract](75)

*+*[PDF](279.9KB)

**Abstract:**

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.

**
**

2016 Impact Factor: 1.099

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