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Discrete & Continuous Dynamical Systems - A
January & February 2008 , Volume 22 , Issue 1&2
A special issue dedicated to Yakov Pesin
on the occasion of his 60th birthday
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2008, 22(1&2): i-i
doi: 10.3934/dcds.2008.22.1i
+[Abstract](1222)
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Abstract:
This special issue of Discrete and Continuous Dynamical Systems is dedicated to Yakov Pesin on the occasion of his sixtieth birthday, which took place in December of 2006. Pesin is one of the world leaders in the field of dynamical systems. His work exerted a deep and lasting influence in the area. Subjects of his landmark works include nonuniform hyperbolicity, smooth ergodic theory, partial hyperbolicity, thermodynamic formalism, and dimension theory in dynamics.
For more information please click the “Full Text” above.
This special issue of Discrete and Continuous Dynamical Systems is dedicated to Yakov Pesin on the occasion of his sixtieth birthday, which took place in December of 2006. Pesin is one of the world leaders in the field of dynamical systems. His work exerted a deep and lasting influence in the area. Subjects of his landmark works include nonuniform hyperbolicity, smooth ergodic theory, partial hyperbolicity, thermodynamic formalism, and dimension theory in dynamics.
For more information please click the “Full Text” above.
2008, 22(1&2): 23-34
doi: 10.3934/dcds.2008.22.23
+[Abstract](1368)
+[PDF](187.9KB)
Abstract:
The $(\epsilon,n)$-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance $\epsilon$ during the time interval $n$. Behavior of the $(\epsilon, n)$-complexity functions as $n\to\infty$ is reflected in the properties of special measures. These measures are constructed as limits of atomic measures supported at points of $(\epsilon,n)$-separated sets. We study such measures. In particular, we prove that they are invariant if the $(\epsilon,n)$-complexity function grows subexponentially.
The $(\epsilon,n)$-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance $\epsilon$ during the time interval $n$. Behavior of the $(\epsilon, n)$-complexity functions as $n\to\infty$ is reflected in the properties of special measures. These measures are constructed as limits of atomic measures supported at points of $(\epsilon,n)$-separated sets. We study such measures. In particular, we prove that they are invariant if the $(\epsilon,n)$-complexity function grows subexponentially.
2008, 22(1&2): 35-61
doi: 10.3934/dcds.2008.22.35
+[Abstract](1760)
+[PDF](319.1KB)
Abstract:
We combine the technique of inducing with a method of Johnson boxes and construct new examples of S-unimodal maps $\varphi$ which do not have a finite absolutely continuous invariant measure, but do have a $\sigma$-finite one which is infinite on every non-trivial interval.
We prove the following dichotomy. Every absolutely continuous invariant measure is either $\sigma$-finite, or else it is infinite on every set of positive Lebesgue measure.
We combine the technique of inducing with a method of Johnson boxes and construct new examples of S-unimodal maps $\varphi$ which do not have a finite absolutely continuous invariant measure, but do have a $\sigma$-finite one which is infinite on every non-trivial interval.
We prove the following dichotomy. Every absolutely continuous invariant measure is either $\sigma$-finite, or else it is infinite on every set of positive Lebesgue measure.
2008, 22(1&2): 63-74
doi: 10.3934/dcds.2008.22.63
+[Abstract](1523)
+[PDF](202.0KB)
Abstract:
In the paper [4] were introduced visual and simple classes of dynamical systems with divided phase space, i.e., with the coexistence of chaotic components of positive measure and islands of regular dynamics. These classes consist of mushroom billiards and their modifications. Here we generalize the construction of mushroom billiards and besides present another class of simple and visual billiards with divided phase space.
In the paper [4] were introduced visual and simple classes of dynamical systems with divided phase space, i.e., with the coexistence of chaotic components of positive measure and islands of regular dynamics. These classes consist of mushroom billiards and their modifications. Here we generalize the construction of mushroom billiards and besides present another class of simple and visual billiards with divided phase space.
2008, 22(1&2): 75-88
doi: 10.3934/dcds.2008.22.75
+[Abstract](1779)
+[PDF](229.5KB)
Abstract:
It is shown that stable accessibility property is $C^r$-dense among partially hyperbolic diffeomorphisms with one-dimensional center bundle, for $r \geq 2$, volume preserving or not. This establishes a conjecture by Pugh and Shub for these systems.
It is shown that stable accessibility property is $C^r$-dense among partially hyperbolic diffeomorphisms with one-dimensional center bundle, for $r \geq 2$, volume preserving or not. This establishes a conjecture by Pugh and Shub for these systems.
2008, 22(1&2): 89-100
doi: 10.3934/dcds.2008.22.89
+[Abstract](1769)
+[PDF](185.4KB)
Abstract:
This paper discusses relationships among the basic notions that have been important in recent investigations of the ergodicity of volume-preserving partially hyperbolic diffeomorphisms. In particular we survey the possible definitions of dynamical coherence and discuss the relationship between dynamical coherence and center bunching.
This paper discusses relationships among the basic notions that have been important in recent investigations of the ergodicity of volume-preserving partially hyperbolic diffeomorphisms. In particular we survey the possible definitions of dynamical coherence and discuss the relationship between dynamical coherence and center bunching.
2008, 22(1&2): 101-109
doi: 10.3934/dcds.2008.22.101
+[Abstract](1798)
+[PDF](151.9KB)
Abstract:
The hard sphere gas is a mathematical model in which several spherical particles collide elastically with each other in a compact Euclidean domain. Using the fact that this system can be modeled as point billiard, its dynamical properties have been investigated extensively, with a great deal of progress towards establishing a central hypothesis, viz., that the system is ergodic. Here we consider the implications of extending the model to include non-spherical particles which have rotational as well as translational components of motion. We show that the point billiard model which forms the basis of the hard sphere gas investigations can be extended to the non-spherical case.
The hard sphere gas is a mathematical model in which several spherical particles collide elastically with each other in a compact Euclidean domain. Using the fact that this system can be modeled as point billiard, its dynamical properties have been investigated extensively, with a great deal of progress towards establishing a central hypothesis, viz., that the system is ergodic. Here we consider the implications of extending the model to include non-spherical particles which have rotational as well as translational components of motion. We show that the point billiard model which forms the basis of the hard sphere gas investigations can be extended to the non-spherical case.
2008, 22(1&2): 111-130
doi: 10.3934/dcds.2008.22.111
+[Abstract](1639)
+[PDF](285.9KB)
Abstract:
Elsewhere [1] we have shown that mechanical systems involving free motion with elastic collisions can be modeled as billiards. In this paper we explore what sorts of billiard systems arise in this way, and we explore some of the dynamical properties that can be determined from modeling the systems as billiards.
Elsewhere [1] we have shown that mechanical systems involving free motion with elastic collisions can be modeled as billiards. In this paper we explore what sorts of billiard systems arise in this way, and we explore some of the dynamical properties that can be determined from modeling the systems as billiards.
2008, 22(1&2): 131-164
doi: 10.3934/dcds.2008.22.131
+[Abstract](2264)
+[PDF](404.0KB)
Abstract:
We introduce a relative Gurevich pressure for random countable topologically mixing Markov shifts. It is shown that the relative variational principle holds for this notion of pressure. We also prove a relative Ruelle-Perron-Frobenius theorem which enables us to construct a wealth of invariant Gibbs measures for locally fiber Hölder continuous functions. This is accomplished via a new construction of an equivariant family of fiber measures using Crauel's relative Prohorov theorem. Some properties of the Gibbs measures are discussed as well.
We introduce a relative Gurevich pressure for random countable topologically mixing Markov shifts. It is shown that the relative variational principle holds for this notion of pressure. We also prove a relative Ruelle-Perron-Frobenius theorem which enables us to construct a wealth of invariant Gibbs measures for locally fiber Hölder continuous functions. This is accomplished via a new construction of an equivariant family of fiber measures using Crauel's relative Prohorov theorem. Some properties of the Gibbs measures are discussed as well.
2008, 22(1&2): 165-182
doi: 10.3934/dcds.2008.22.165
+[Abstract](1738)
+[PDF](241.7KB)
Abstract:
We consider a ball bouncing off infinitely heavy periodically moving plate in the presence of a potential force. Assuming that the potential equals to a power of the ball's height we present conditions guaranteeing recurrence in the sense that the total energy of almost every trajectory does not go to infinity.
We consider a ball bouncing off infinitely heavy periodically moving plate in the presence of a potential force. Assuming that the potential equals to a power of the ball's height we present conditions guaranteeing recurrence in the sense that the total energy of almost every trajectory does not go to infinity.
2008, 22(1&2): 183-200
doi: 10.3934/dcds.2008.22.183
+[Abstract](1857)
+[PDF](255.2KB)
Abstract:
We consider two $C^2$ Anosov diffeomorphisms in a $C^1$ neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum. We prove that they are $C^{1+\nu}$ conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues.
We consider two $C^2$ Anosov diffeomorphisms in a $C^1$ neighborhood of a linear hyperbolic automorphism of three dimensional torus with real spectrum. We prove that they are $C^{1+\nu}$ conjugate if and only if the differentials of the return maps at corresponding periodic points have the same eigenvalues.
2008, 22(1&2): 201-213
doi: 10.3934/dcds.2008.22.201
+[Abstract](2185)
+[PDF](211.4KB)
Abstract:
The literature contains several extensions of the standard definitions of topological entropy for a continuous self-map $f: X \rightarrow X$ from the case when $X$ is a compact metric space to the case when $X$ is allowed to be noncompact. These extensions all require the space $X$ to be totally bounded, or equivalently to have a compact completion, and are invariants of uniform conjugacy. When the map $f$ is uniformly continuous, it extends continuously to the completion, and the various notions of entropy reduce to the standard ones (applied to this extension). However, when uniform continuity is not assumed, these new quantities can differ. We consider extensions proposed by Bowen (maximizing over compact subsets and a definition of Hausdorff dimension type) and Friedland (using the compactification of the graph of $f$) as well as a straightforward extension of Bowen and Dinaburg's definition from the compact case, assuming that $X$ is totally bounded, but not necessarily compact. This last extension agrees with Friedland's, and both dominate the one proposed by Bowen (Theorem 6). Examples show how varying the metric outside its uniform class can vary both quantities. The natural extension of Adler--Konheim--McAndrew's original (metric-free) definition of topological entropy beyond compact spaces dominates these other notions, and is unfortunately infinite for a great number of noncompact examples.
The literature contains several extensions of the standard definitions of topological entropy for a continuous self-map $f: X \rightarrow X$ from the case when $X$ is a compact metric space to the case when $X$ is allowed to be noncompact. These extensions all require the space $X$ to be totally bounded, or equivalently to have a compact completion, and are invariants of uniform conjugacy. When the map $f$ is uniformly continuous, it extends continuously to the completion, and the various notions of entropy reduce to the standard ones (applied to this extension). However, when uniform continuity is not assumed, these new quantities can differ. We consider extensions proposed by Bowen (maximizing over compact subsets and a definition of Hausdorff dimension type) and Friedland (using the compactification of the graph of $f$) as well as a straightforward extension of Bowen and Dinaburg's definition from the compact case, assuming that $X$ is totally bounded, but not necessarily compact. This last extension agrees with Friedland's, and both dominate the one proposed by Bowen (Theorem 6). Examples show how varying the metric outside its uniform class can vary both quantities. The natural extension of Adler--Konheim--McAndrew's original (metric-free) definition of topological entropy beyond compact spaces dominates these other notions, and is unfortunately infinite for a great number of noncompact examples.
2008, 22(1&2): 215-234
doi: 10.3934/dcds.2008.22.215
+[Abstract](1957)
+[PDF](267.6KB)
Abstract:
Let $M^{n}$ be a compact $C^{\infty}$ Riemannian manifold of dimension $n\geq 2$. Let $\text{Diff}^{\r }(M^{n})$ be the space of all $C^{\r }$ diffeomorphisms of $M^{n}$, where $ 1 < r \le \infty$. For a $C^{\r }$ diffeomorphism $f$ in $\text{Diff}^{\r }(M^{n})$ with a hyperbolic attractor $\Lambda_{f}$ on which $f$ is topologically transitive, let $U(f)$ be the $C^{1}$ open set of $\text{Diff}^{\r }(M^{n})$ such that each element in $U(f)$ can be connected to $f$ by finitely many $C^{1}$ structural stability balls in $\text{Diff}^{\r }(M^{n})$. Then by the structural stability, any element $g$ in $U(f)$ has a hyperbolic attractor $\Lambda_{g}$ and $g|\Lambda_{g}$ is topologically conjugate to $f|\Lambda_{f}$. Therefore, the topological entropy $h(g|\Lambda_{g})$ is a constant function when it is restricted to $U(f)$. However, the metric entropy $h_{\mu}(g)$ with respect to the SRB measure $\mu=\mu_{g}$ can vary. We prove that the infimum of the metric entropy $h_{\mu}(g)$ on $U(f)$ is zero.
Let $M^{n}$ be a compact $C^{\infty}$ Riemannian manifold of dimension $n\geq 2$. Let $\text{Diff}^{\r }(M^{n})$ be the space of all $C^{\r }$ diffeomorphisms of $M^{n}$, where $ 1 < r \le \infty$. For a $C^{\r }$ diffeomorphism $f$ in $\text{Diff}^{\r }(M^{n})$ with a hyperbolic attractor $\Lambda_{f}$ on which $f$ is topologically transitive, let $U(f)$ be the $C^{1}$ open set of $\text{Diff}^{\r }(M^{n})$ such that each element in $U(f)$ can be connected to $f$ by finitely many $C^{1}$ structural stability balls in $\text{Diff}^{\r }(M^{n})$. Then by the structural stability, any element $g$ in $U(f)$ has a hyperbolic attractor $\Lambda_{g}$ and $g|\Lambda_{g}$ is topologically conjugate to $f|\Lambda_{f}$. Therefore, the topological entropy $h(g|\Lambda_{g})$ is a constant function when it is restricted to $U(f)$. However, the metric entropy $h_{\mu}(g)$ with respect to the SRB measure $\mu=\mu_{g}$ can vary. We prove that the infimum of the metric entropy $h_{\mu}(g)$ on $U(f)$ is zero.
2008, 22(1&2): 235-246
doi: 10.3934/dcds.2008.22.235
+[Abstract](1968)
+[PDF](1088.9KB)
Abstract:
In this note we consider measures supported on limit sets of systems that contract on average. In particular, we present an upper bound on their Hausdorff dimension.
In this note we consider measures supported on limit sets of systems that contract on average. In particular, we present an upper bound on their Hausdorff dimension.
2008, 22(1&2): 247-325
doi: 10.3934/dcds.2008.22.247
+[Abstract](1757)
+[PDF](806.5KB)
Abstract:
We study the almost sure asymptotic behavior of the ergodic sums of $L^1$--functions, for the infinite measure preserving system given by the horocycle flow on the unit tangent bundle of a $\Z^d$--cover of a hyperbolic surface of finite area, equipped with the volume measure. We prove rational ergodicity, identify the return sequence, and describe the fluctuations of the ergodic sums normalized by the return sequence. One application is a 'second order ergodic theorem': almost sure convergence of properly normalized ergodic sums, subject to a certain summability method (the ordinary pointwise ergodic theorem fails for infinite measure preserving systems).
We study the almost sure asymptotic behavior of the ergodic sums of $L^1$--functions, for the infinite measure preserving system given by the horocycle flow on the unit tangent bundle of a $\Z^d$--cover of a hyperbolic surface of finite area, equipped with the volume measure. We prove rational ergodicity, identify the return sequence, and describe the fluctuations of the ergodic sums normalized by the return sequence. One application is a 'second order ergodic theorem': almost sure convergence of properly normalized ergodic sums, subject to a certain summability method (the ordinary pointwise ergodic theorem fails for infinite measure preserving systems).
2008, 22(1&2): 327-344
doi: 10.3934/dcds.2008.22.327
+[Abstract](1444)
+[PDF](260.1KB)
Abstract:
For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition.
For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition.
2008, 22(1&2): 345-412
doi: 10.3934/dcds.2008.22.345
+[Abstract](1437)
+[PDF](532.2KB)
Abstract:
We define new distortion quantities for diffeomorphisms of the Euclidean plane and study their properties. In particular, we obtain composition rules for these quantities analogous to standard rules for maps of an interval. Our results apply to maps with unbounded derivatives and have important applications in the theory of SRB measures for surface diffeomorphisms.
We define new distortion quantities for diffeomorphisms of the Euclidean plane and study their properties. In particular, we obtain composition rules for these quantities analogous to standard rules for maps of an interval. Our results apply to maps with unbounded derivatives and have important applications in the theory of SRB measures for surface diffeomorphisms.
2008, 22(1&2): 413-425
doi: 10.3934/dcds.2008.22.413
+[Abstract](1603)
+[PDF](201.4KB)
Abstract:
Let $U$ be an open set in $\mathbb R^2$ and $f:U\to \mathbb R$ a function. $f$ is said to be $C^{n,\alpha}$ if it is $C^n$ and has the $n$-th derivative $\alpha$-Hölder, $0<\alpha< 1$. We generalize a result due to Journé [2] about the $C^{n,\alpha}$ regularity of a real valued continuous function on $U$ that is $C^{n,\alpha}$ along two transverse continuous foliations with $C^{n,\alpha}$ leaves. For $\omega$ a Dini modulus of continuity, $f$ is said to be $C^{n,\omega}$ if it is $C^n$ and has the $n$-th derivative bounded in the seminorm defined by $\omega$. We assume that $f$ is $C^{n,\omega}$ along two transverse continuous foliations with $C^{n,\omega}$ leaves, and show that, under an additional summability condition for the modulus, $f$ is $C^{n,\omega'}$ for $\omega'(t)=\int^t_0\frac{\omega(\tau)}{\tau}d\tau.$ For $\omega(t)=t^{\alpha}, 0<\alpha< 1,$ one recovers Journé's result. Examples of moduli that satisfy our assumptions are given by $\omega(t)=t^{\alpha}( \ln\frac{1}{t} )^{\beta}$, for $0<\alpha<1$ and $0<\beta$.
Let $U$ be an open set in $\mathbb R^2$ and $f:U\to \mathbb R$ a function. $f$ is said to be $C^{n,\alpha}$ if it is $C^n$ and has the $n$-th derivative $\alpha$-Hölder, $0<\alpha< 1$. We generalize a result due to Journé [2] about the $C^{n,\alpha}$ regularity of a real valued continuous function on $U$ that is $C^{n,\alpha}$ along two transverse continuous foliations with $C^{n,\alpha}$ leaves. For $\omega$ a Dini modulus of continuity, $f$ is said to be $C^{n,\omega}$ if it is $C^n$ and has the $n$-th derivative bounded in the seminorm defined by $\omega$. We assume that $f$ is $C^{n,\omega}$ along two transverse continuous foliations with $C^{n,\omega}$ leaves, and show that, under an additional summability condition for the modulus, $f$ is $C^{n,\omega'}$ for $\omega'(t)=\int^t_0\frac{\omega(\tau)}{\tau}d\tau.$ For $\omega(t)=t^{\alpha}, 0<\alpha< 1,$ one recovers Journé's result. Examples of moduli that satisfy our assumptions are given by $\omega(t)=t^{\alpha}( \ln\frac{1}{t} )^{\beta}$, for $0<\alpha<1$ and $0<\beta$.
2008, 22(1&2): 427-443
doi: 10.3934/dcds.2008.22.427
+[Abstract](1711)
+[PDF](252.0KB)
Abstract:
For the study of hard ball systems, the algebro-geometric approach appeared in 1999 --- in a sense surprisingly but quite efficiently --- for proving the hyperbolicity of typical systems (see [26]). An improvement by Simányi [22] also provided the ergodicity of typical systems, thus an almost complete proof of the Boltzmann--Sinai ergodic hypothesis. More than that, at present, the best form of the local ergodicity theorem for semi-dispersing billiards, [6] also uses algebraic methods (and the algebraicity condition on the scatterers). The goal of the present paper is to discuss the essential steps of the algebro-geometric approach by assuming and using possibly minimum information about hard ball systems. In particular, we also minimize the intersection of the material with the earlier surveys [29] and [20].
For the study of hard ball systems, the algebro-geometric approach appeared in 1999 --- in a sense surprisingly but quite efficiently --- for proving the hyperbolicity of typical systems (see [26]). An improvement by Simányi [22] also provided the ergodicity of typical systems, thus an almost complete proof of the Boltzmann--Sinai ergodic hypothesis. More than that, at present, the best form of the local ergodicity theorem for semi-dispersing billiards, [6] also uses algebraic methods (and the algebraicity condition on the scatterers). The goal of the present paper is to discuss the essential steps of the algebro-geometric approach by assuming and using possibly minimum information about hard ball systems. In particular, we also minimize the intersection of the material with the earlier surveys [29] and [20].
2008, 22(1&2): 445-464
doi: 10.3934/dcds.2008.22.445
+[Abstract](1720)
+[PDF](237.4KB)
Abstract:
We shall consider piecewise invertible sofic systems admitting indifferent periodic orbits and establish polynomial lower bounds on the decay of correlations associated with weak Gibbs measures.
We shall consider piecewise invertible sofic systems admitting indifferent periodic orbits and establish polynomial lower bounds on the decay of correlations associated with weak Gibbs measures.
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