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Discrete & Continuous Dynamical Systems - A
September 2013 , Volume 33 , Issue 9
Regular Papers: 3835-4205; Special Issue Papers: 4207-4347
Special Issue Papers are related to the satellite conference on
"Various Aspects of Dynamical Systems," following ICM 2010
Guest Editors: Jon Aaronson, Christian Bonatti, S. G. Dani and Tarun Das
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2013, 33(9): 3835-3860
doi: 10.3934/dcds.2013.33.3835
+[Abstract](2307)
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Abstract:
Oseledets' celebrated Multiplicative Ergodic Theorem (MET) [V.I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179--210.] is concerned with the exponential growth rates of vectors under the action of a linear cocycle on $\mathbb{R}^d$. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of $\mathbb{R}^d$ into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated [G. Froyland, S. Lloyd, and A. Quas, Coherent structures and exceptional spectrum for Perron--Frobenius cocycles, Ergodic Theory and Dynam. Systems 30 (2010), , 729--756.] that a splitting over $\mathbb{R}^d$ is guaranteed without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.
Oseledets' celebrated Multiplicative Ergodic Theorem (MET) [V.I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179--210.] is concerned with the exponential growth rates of vectors under the action of a linear cocycle on $\mathbb{R}^d$. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of $\mathbb{R}^d$ into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated [G. Froyland, S. Lloyd, and A. Quas, Coherent structures and exceptional spectrum for Perron--Frobenius cocycles, Ergodic Theory and Dynam. Systems 30 (2010), , 729--756.] that a splitting over $\mathbb{R}^d$ is guaranteed without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.
2013, 33(9): 3861-3884
doi: 10.3934/dcds.2013.33.3861
+[Abstract](2152)
+[PDF](469.7KB)
Abstract:
We study the initial boundary value problem for the Schrödinger equation with non-homogeneous Dirichlet boundary conditions. Special care is devoted to the space where the boundary data belong. When $\Omega$ is the complement of a non-trapping obstacle, well-posedness for boundary data of optimal regularity is obtained by transposition arguments. If $\Omega^c$ is convex, a local smoothing property (similar to the one for the Cauchy problem) is proved, and used to obtain Strichartz estimates. As an application local well-posedness for a class of subcritical non-linear Schrödinger equations is derived.
We study the initial boundary value problem for the Schrödinger equation with non-homogeneous Dirichlet boundary conditions. Special care is devoted to the space where the boundary data belong. When $\Omega$ is the complement of a non-trapping obstacle, well-posedness for boundary data of optimal regularity is obtained by transposition arguments. If $\Omega^c$ is convex, a local smoothing property (similar to the one for the Cauchy problem) is proved, and used to obtain Strichartz estimates. As an application local well-posedness for a class of subcritical non-linear Schrödinger equations is derived.
2013, 33(9): 3885-3901
doi: 10.3934/dcds.2013.33.3885
+[Abstract](1910)
+[PDF](672.6KB)
Abstract:
Absolutely continuous invariant measures (acims) for general induced transformations are shown to be related, in a natural way, to popular tower constructions regardless of any particulars of the latter. When combined with (an appropriate generalization of) the known integrability criterion for the existence of such acims, this leads to necessary and sufficient conditions under which acims can be lifted to, or projected from, nonsingular extensions.
Absolutely continuous invariant measures (acims) for general induced transformations are shown to be related, in a natural way, to popular tower constructions regardless of any particulars of the latter. When combined with (an appropriate generalization of) the known integrability criterion for the existence of such acims, this leads to necessary and sufficient conditions under which acims can be lifted to, or projected from, nonsingular extensions.
2013, 33(9): 3903-3913
doi: 10.3934/dcds.2013.33.3903
+[Abstract](1670)
+[PDF](313.3KB)
Abstract:
This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. I prove that the only convex billiard without conjugate points on the hyperbolic plane or on the hemisphere is a circular billiard.
This paper deals with Hopf type rigidity for convex billiards on surfaces of constant curvature. I prove that the only convex billiard without conjugate points on the hyperbolic plane or on the hemisphere is a circular billiard.
2013, 33(9): 3915-3936
doi: 10.3934/dcds.2013.33.3915
+[Abstract](2448)
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Abstract:
This paper is mainly devoted to the study of the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. We consider the case when the two zones are separated by a straight line $\Sigma$ and the singular point of the unperturbed system is in $\Sigma$. It is proved that the maximum number of limit cycles that can appear up to a seventh order perturbation is three. Moreover this upper bound is reached. This result confirms that these systems have more limit cycles than it was expected. Finally, center and isochronicity problems are also studied in systems which include a first order perturbation. For the latter systems it is also proved that, when the period function, defined in the period annulus of the center, is not monotone, then it has at most one critical period. Moreover this upper bound is also reached.
This paper is mainly devoted to the study of the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. We consider the case when the two zones are separated by a straight line $\Sigma$ and the singular point of the unperturbed system is in $\Sigma$. It is proved that the maximum number of limit cycles that can appear up to a seventh order perturbation is three. Moreover this upper bound is reached. This result confirms that these systems have more limit cycles than it was expected. Finally, center and isochronicity problems are also studied in systems which include a first order perturbation. For the latter systems it is also proved that, when the period function, defined in the period annulus of the center, is not monotone, then it has at most one critical period. Moreover this upper bound is also reached.
2013, 33(9): 3937-3955
doi: 10.3934/dcds.2013.33.3937
+[Abstract](2512)
+[PDF](466.1KB)
Abstract:
In this paper we consider the following semi-linear poly-harmonic equation with Navier boundary conditions on the half space $R^n_+$: \begin{equation} \left\{\begin{array}{l} (-\triangle)^{\frac{\alpha}{2}} u=u^p,\ \ \ \ \ \:\:\: \:\:\:\:\:\ \:\:\ \ \ \ \ \ \ \ \ \ \ \ \:\:\:\:\ \mbox{in}\,\ R^n_+,\\ u=-\triangle u=\cdots=(-\triangle)^{\frac{\alpha}{2}-1}u=0, \ \ \ \mbox{on}\ \partial R^n_+, \end{array} \right. \label{phe1} \end{equation} where $\alpha$ is any even number between $0$ and $n$, and $p>1$.
First we prove that (1) is equivalent to the following integral equation \begin{equation} u(x)=\int_{R^n_+}G(x,y,\alpha) u^p(y)dy,\,\,\,\,\, x\in\,R^n_+, \label{ie0} \end{equation} under some very mild growth condition, where $G(x, y,\alpha)$ is the Green's function associated with the same Navier boundary conditions on the half-space .
Then by combining the method of moving planes in integral forms with a certain type of Kelvin transform, we obtain the non-existence of positive solutions for integral equation (2) in both subcritical and critical cases under only local integrability conditions. This remarkably weaken the global integrability assumptions on solutions in paper [3]. Our results on integral equation (2) are valid for all real values $\alpha$ between $0$ and $n$.
Finally, we establish a Liouville type theorem for PDE (1), and this generalizes Guo and Liu's result [21] by significantly weaken the growth conditions on the solutions.
In this paper we consider the following semi-linear poly-harmonic equation with Navier boundary conditions on the half space $R^n_+$: \begin{equation} \left\{\begin{array}{l} (-\triangle)^{\frac{\alpha}{2}} u=u^p,\ \ \ \ \ \:\:\: \:\:\:\:\:\ \:\:\ \ \ \ \ \ \ \ \ \ \ \ \:\:\:\:\ \mbox{in}\,\ R^n_+,\\ u=-\triangle u=\cdots=(-\triangle)^{\frac{\alpha}{2}-1}u=0, \ \ \ \mbox{on}\ \partial R^n_+, \end{array} \right. \label{phe1} \end{equation} where $\alpha$ is any even number between $0$ and $n$, and $p>1$.
First we prove that (1) is equivalent to the following integral equation \begin{equation} u(x)=\int_{R^n_+}G(x,y,\alpha) u^p(y)dy,\,\,\,\,\, x\in\,R^n_+, \label{ie0} \end{equation} under some very mild growth condition, where $G(x, y,\alpha)$ is the Green's function associated with the same Navier boundary conditions on the half-space .
Then by combining the method of moving planes in integral forms with a certain type of Kelvin transform, we obtain the non-existence of positive solutions for integral equation (2) in both subcritical and critical cases under only local integrability conditions. This remarkably weaken the global integrability assumptions on solutions in paper [3]. Our results on integral equation (2) are valid for all real values $\alpha$ between $0$ and $n$.
Finally, we establish a Liouville type theorem for PDE (1), and this generalizes Guo and Liu's result [21] by significantly weaken the growth conditions on the solutions.
2013, 33(9): 3957-3980
doi: 10.3934/dcds.2013.33.3957
+[Abstract](2354)
+[PDF](544.4KB)
Abstract:
This article is concerned with conjugacy problems arising in the homeomorphisms group, Hom($F$), of unbounded subsets $F$ of normed vector spaces $E$. Given two homeomorphisms $f$ and $g$ in Hom($F$), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom($F$), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.
This article is concerned with conjugacy problems arising in the homeomorphisms group, Hom($F$), of unbounded subsets $F$ of normed vector spaces $E$. Given two homeomorphisms $f$ and $g$ in Hom($F$), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom($F$), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.
2013, 33(9): 3981-4002
doi: 10.3934/dcds.2013.33.3981
+[Abstract](1754)
+[PDF](480.6KB)
Abstract:
Let $( Ω , Α , \mathbb{P} , \tau )$ be an ergodic dynamical system. The rotated ergodic sums of a function $f$ on $\Omega$ for $\theta \in \mathbb{R}$ are $S_n^θ f : = \sum_{k=0}^{n-1} e^{2\pi i k \theta} f \circ \tau^k, n \geq 1$. Using Carleson's theorem on Fourier series, Peligrad and Wu proved in [14] that $(S_n^\theta f)_{n \geq 1}$ satisfies the CLT for a.e. $\theta$ when $(f\circ \tau^n)$ is a regular process.
Our aim is to extend this result and give a simple proof based on the Fejér-Lebesgue theorem. The results are expressed in the framework of processes generated by $K$-systems. We also consider the invariance principle for modified rotated sums. In a last section, we extend the method to $\mathbb{Z}^d$-dynamical systems.
Let $( Ω , Α , \mathbb{P} , \tau )$ be an ergodic dynamical system. The rotated ergodic sums of a function $f$ on $\Omega$ for $\theta \in \mathbb{R}$ are $S_n^θ f : = \sum_{k=0}^{n-1} e^{2\pi i k \theta} f \circ \tau^k, n \geq 1$. Using Carleson's theorem on Fourier series, Peligrad and Wu proved in [14] that $(S_n^\theta f)_{n \geq 1}$ satisfies the CLT for a.e. $\theta$ when $(f\circ \tau^n)$ is a regular process.
Our aim is to extend this result and give a simple proof based on the Fejér-Lebesgue theorem. The results are expressed in the framework of processes generated by $K$-systems. We also consider the invariance principle for modified rotated sums. In a last section, we extend the method to $\mathbb{Z}^d$-dynamical systems.
2013, 33(9): 4003-4015
doi: 10.3934/dcds.2013.33.4003
+[Abstract](1838)
+[PDF](407.5KB)
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We study equilibrium behavior for two-sided topological Markov shifts with a countable number of states. We assume the associated potential is Walters with finite first variation and that the shift is topologically transitive. We show the resulting equilibrium measure is Bernoulli up to a period.
We study equilibrium behavior for two-sided topological Markov shifts with a countable number of states. We assume the associated potential is Walters with finite first variation and that the shift is topologically transitive. We show the resulting equilibrium measure is Bernoulli up to a period.
2013, 33(9): 4017-4040
doi: 10.3934/dcds.2013.33.4017
+[Abstract](2013)
+[PDF](1088.2KB)
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We consider the motion of a planar rigid body in a potential two-dimensional flow with a circulation and subject to a certain nonholonomic constraint. This model can be related to the design of underwater vehicles.
The equations of motion admit a reduction to a 2-dimensional nonlinear system, which is integrated explicitly. We show that the reduced system comprises both asymptotic and periodic dynamics separated by a critical value of the energy, and give a complete classification of types of the motion. Then we describe the whole variety of the trajectories of the body on the plane.
We consider the motion of a planar rigid body in a potential two-dimensional flow with a circulation and subject to a certain nonholonomic constraint. This model can be related to the design of underwater vehicles.
The equations of motion admit a reduction to a 2-dimensional nonlinear system, which is integrated explicitly. We show that the reduced system comprises both asymptotic and periodic dynamics separated by a critical value of the energy, and give a complete classification of types of the motion. Then we describe the whole variety of the trajectories of the body on the plane.
Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction
2013, 33(9): 4041-4069
doi: 10.3934/dcds.2013.33.4041
+[Abstract](2425)
+[PDF](494.3KB)
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We consider the reaction-diffusion system $u_t=\delta u_{xx}-2uv/(\beta+u)$, $v_t=v_{xx}+uv/(\beta+u)$, which is used to model the acidic nitrate-ferroin reaction. Here $\beta$ is a positive constant, $u$ and $v$ represent the concentrations of the ferroin and acidic nitrate respectively, and $\delta$ denotes the ratio of the diffusion rates. The existence of travelling waves for this system is known. Using energy functionals, we provide a stability analysis of travelling waves.
We consider the reaction-diffusion system $u_t=\delta u_{xx}-2uv/(\beta+u)$, $v_t=v_{xx}+uv/(\beta+u)$, which is used to model the acidic nitrate-ferroin reaction. Here $\beta$ is a positive constant, $u$ and $v$ represent the concentrations of the ferroin and acidic nitrate respectively, and $\delta$ denotes the ratio of the diffusion rates. The existence of travelling waves for this system is known. Using energy functionals, we provide a stability analysis of travelling waves.
2013, 33(9): 4071-4093
doi: 10.3934/dcds.2013.33.4071
+[Abstract](1997)
+[PDF](7569.5KB)
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A notion of global attraction and repulsion of heteroclinic limit cycles is introduced for strongly competitive Kolmogorov systems. Conditions are obtained for the existence of cycles linking the full set of axial equilibria and their global asymptotic behaviour on the carrying simplex. The global dynamics of systems with a heteroclinic limit cycle is studied. Results are also obtained for Kolmogorov systems where some components vanish as $t\rightarrow \pm \infty$.
A notion of global attraction and repulsion of heteroclinic limit cycles is introduced for strongly competitive Kolmogorov systems. Conditions are obtained for the existence of cycles linking the full set of axial equilibria and their global asymptotic behaviour on the carrying simplex. The global dynamics of systems with a heteroclinic limit cycle is studied. Results are also obtained for Kolmogorov systems where some components vanish as $t\rightarrow \pm \infty$.
2013, 33(9): 4095-4122
doi: 10.3934/dcds.2013.33.4095
+[Abstract](1811)
+[PDF](613.0KB)
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The initial value problem of the Zakharov system on a two-dimensional torus with general period is considered in this paper. We apply the $I$-method with some `resonant decomposition' to show global well-posedness results for small-in-$L^2$ initial data belonging to some spaces weaker than the energy class. We also consider an application of our ideas to the initial value problem on $\mathbb{R}^2$ and give an improvement of the best known result by Pecher (2012).
The initial value problem of the Zakharov system on a two-dimensional torus with general period is considered in this paper. We apply the $I$-method with some `resonant decomposition' to show global well-posedness results for small-in-$L^2$ initial data belonging to some spaces weaker than the energy class. We also consider an application of our ideas to the initial value problem on $\mathbb{R}^2$ and give an improvement of the best known result by Pecher (2012).
2013, 33(9): 4123-4155
doi: 10.3934/dcds.2013.33.4123
+[Abstract](2564)
+[PDF](557.2KB)
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Consider a random cocycle $\Phi$ on a separable infinite-dimensional Hilbert space preserving a probability measure $\mu$, which is supported on a random compact set $K$. We show that if $\Phi$ is $C^2$ (over $K$) and satisfies some mild integrable conditions of the differentials, then Pesin's entropy formula holds if $\mu$ has absolutely continuous conditional measures on the unstable manifolds. The converse is also true under an additional condition on $K$ when the system has no zero Lyapunov exponent.
Consider a random cocycle $\Phi$ on a separable infinite-dimensional Hilbert space preserving a probability measure $\mu$, which is supported on a random compact set $K$. We show that if $\Phi$ is $C^2$ (over $K$) and satisfies some mild integrable conditions of the differentials, then Pesin's entropy formula holds if $\mu$ has absolutely continuous conditional measures on the unstable manifolds. The converse is also true under an additional condition on $K$ when the system has no zero Lyapunov exponent.
2013, 33(9): 4157-4171
doi: 10.3934/dcds.2013.33.4157
+[Abstract](1751)
+[PDF](417.3KB)
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In this article, we continue our study of category dynamical systems, that is functors $s$ from a category $G$ to $Top^{op}$, and their corresponding skew category algebras. Suppose that the spaces $s(e)$, for $e ∈ ob(G)$, are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) $G$ is inverse connected, (iii) $s$ is minimal and (iv) $s$ is faithful. We also show that if $G$ is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by Öinert for skew group algebras to a large class of skew category algebras.
In this article, we continue our study of category dynamical systems, that is functors $s$ from a category $G$ to $Top^{op}$, and their corresponding skew category algebras. Suppose that the spaces $s(e)$, for $e ∈ ob(G)$, are compact Hausdorff. We show that if (i) the skew category algebra is simple, then (ii) $G$ is inverse connected, (iii) $s$ is minimal and (iv) $s$ is faithful. We also show that if $G$ is a locally abelian groupoid, then (i) is equivalent to (ii), (iii) and (iv). Thereby, we generalize results by Öinert for skew group algebras to a large class of skew category algebras.
2013, 33(9): 4173-4186
doi: 10.3934/dcds.2013.33.4173
+[Abstract](1755)
+[PDF](404.8KB)
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When looking at a dynamical system, a natural question to ask is what are its endomorphisms. Using Coven's work in [1] on the endomorphisms of dynamical systems generated by substitutions of equal length on {0,1} as a guide, we fully describe the endomorphisms for a class of almost automorphic symbolic dynamical systems provided there are certain conditions on the set where the factor map fails to be 1-1. While this result does have conditions on both the dynamical system and the factor map, it applies to Sturmian systems and generalized Sturmian systems. We also prove a similar result for a particular 2-dimensional system with a $\mathbb{Z}^2$-action, the discrete chair substitution tiling system.
When looking at a dynamical system, a natural question to ask is what are its endomorphisms. Using Coven's work in [1] on the endomorphisms of dynamical systems generated by substitutions of equal length on {0,1} as a guide, we fully describe the endomorphisms for a class of almost automorphic symbolic dynamical systems provided there are certain conditions on the set where the factor map fails to be 1-1. While this result does have conditions on both the dynamical system and the factor map, it applies to Sturmian systems and generalized Sturmian systems. We also prove a similar result for a particular 2-dimensional system with a $\mathbb{Z}^2$-action, the discrete chair substitution tiling system.
Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory
2013, 33(9): 4187-4205
doi: 10.3934/dcds.2013.33.4187
+[Abstract](2507)
+[PDF](404.3KB)
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We generalize two classical results of Maizel and Pliss that describe relations between hyperbolicity properties of linear system of difference equations and its ability to have a bounded solution for every bounded inhomogeneity. We also apply one of this generalizations in shadowing theory of diffeomorphisms to prove that some sort of limit shadowing is equivalent to structural stability.
We generalize two classical results of Maizel and Pliss that describe relations between hyperbolicity properties of linear system of difference equations and its ability to have a bounded solution for every bounded inhomogeneity. We also apply one of this generalizations in shadowing theory of diffeomorphisms to prove that some sort of limit shadowing is equivalent to structural stability.
2013, 33(9): 4207-4232
doi: 10.3934/dcds.2013.33.4207
+[Abstract](2751)
+[PDF](487.6KB)
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Spacing subshifts were introduced by Lau and Zame in 1973 to provide accessible examples of maps that are (topologically) weakly mixing but not mixing. Although they show a rich variety of dynamical characteristics, they have received little subsequent attention in the dynamical systems literature. This paper is a systematic study of their dynamical properties and shows that they may be used to provide examples of dynamical systems with a huge range of interesting dynamical behaviors. In a later paper we propose to consider in more detail the case when spacing subshifts are also sofic and transitive.
Spacing subshifts were introduced by Lau and Zame in 1973 to provide accessible examples of maps that are (topologically) weakly mixing but not mixing. Although they show a rich variety of dynamical characteristics, they have received little subsequent attention in the dynamical systems literature. This paper is a systematic study of their dynamical properties and shows that they may be used to provide examples of dynamical systems with a huge range of interesting dynamical behaviors. In a later paper we propose to consider in more detail the case when spacing subshifts are also sofic and transitive.
2013, 33(9): 4233-4237
doi: 10.3934/dcds.2013.33.4233
+[Abstract](1764)
+[PDF](310.4KB)
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We characterize the sets of periodic points of bounded linear operators on a Hilbert space $H$. We also find the pairs $(A,M)$, where $A \subset \mathbb{N}$, $M \subset H$ such that there exists a bounded linear operator $T$ on $H$ with $A$ as the set of periods and $M$ as the set of periodic points.
We characterize the sets of periodic points of bounded linear operators on a Hilbert space $H$. We also find the pairs $(A,M)$, where $A \subset \mathbb{N}$, $M \subset H$ such that there exists a bounded linear operator $T$ on $H$ with $A$ as the set of periods and $M$ as the set of periodic points.
2013, 33(9): 4239-4269
doi: 10.3934/dcds.2013.33.4239
+[Abstract](2022)
+[PDF](550.1KB)
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Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$ be a countable group of $\nu$-preserving invertible maps of $X$ into itself. To a probability measure $\mu$ on $\Gamma$ corresponds a random walk on $X$ with Markov operator $P$ given by $P\psi(x) = \sum_{a} \psi(ax) \, \mu(a)$. We consider various examples of ergodic $\Gamma$-actions and random walks and their extensions by a vector space: groups of automorphisms or affine transformations on compact nilmanifolds, random walks in random scenery on non amenable groups, translations on homogeneous spaces of simple Lie groups, random walks on motion groups. A powerful tool in this study is the spectral gap property for the operator $P$ when it holds. We use it to obtain limit theorems, recurrence/transience property and ergodicity for random walks on non compact extensions of the corresponding dynamical systems.
Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$ be a countable group of $\nu$-preserving invertible maps of $X$ into itself. To a probability measure $\mu$ on $\Gamma$ corresponds a random walk on $X$ with Markov operator $P$ given by $P\psi(x) = \sum_{a} \psi(ax) \, \mu(a)$. We consider various examples of ergodic $\Gamma$-actions and random walks and their extensions by a vector space: groups of automorphisms or affine transformations on compact nilmanifolds, random walks in random scenery on non amenable groups, translations on homogeneous spaces of simple Lie groups, random walks on motion groups. A powerful tool in this study is the spectral gap property for the operator $P$ when it holds. We use it to obtain limit theorems, recurrence/transience property and ergodicity for random walks on non compact extensions of the corresponding dynamical systems.
2013, 33(9): 4271-4289
doi: 10.3934/dcds.2013.33.4271
+[Abstract](1817)
+[PDF](575.7KB)
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Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by $T$) is $E$. Moreover, for each non-zero $t\in\Bbb R$, the set of spectral multiplicities of the transformation $T_t$ is also $E$. These results are partly extended to actions of some other locally compact second countable Abelian groups.
Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by $T$) is $E$. Moreover, for each non-zero $t\in\Bbb R$, the set of spectral multiplicities of the transformation $T_t$ is also $E$. These results are partly extended to actions of some other locally compact second countable Abelian groups.
2013, 33(9): 4291-4303
doi: 10.3934/dcds.2013.33.4291
+[Abstract](1818)
+[PDF](956.7KB)
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The striking boundary dependency, the Arctic Circle Phenomenon, exhibited in the Ice model on the square lattice extends to other planar set-ups. This can be shown using a dynamical formulation which we present for the Archimedean lattices. Critical connectivity results guarantee that the Ice configurations can be generated using the simplest and most efficient local actions. Height functions are utilized throughout the analysis. On a hexagon with suitable boundary height the cellular automaton dynamics generates highly nontrivial Ice equilibria in the triangular and Kagomé cases. On the remaining Archimedean lattice for which the Ice model can be defined, the 3.4.6.4. lattice, the long range behavior is shown to be completely different due to strictly positive entropy for all boundary conditions.
The striking boundary dependency, the Arctic Circle Phenomenon, exhibited in the Ice model on the square lattice extends to other planar set-ups. This can be shown using a dynamical formulation which we present for the Archimedean lattices. Critical connectivity results guarantee that the Ice configurations can be generated using the simplest and most efficient local actions. Height functions are utilized throughout the analysis. On a hexagon with suitable boundary height the cellular automaton dynamics generates highly nontrivial Ice equilibria in the triangular and Kagomé cases. On the remaining Archimedean lattice for which the Ice model can be defined, the 3.4.6.4. lattice, the long range behavior is shown to be completely different due to strictly positive entropy for all boundary conditions.
2013, 33(9): 4305-4321
doi: 10.3934/dcds.2013.33.4305
+[Abstract](1586)
+[PDF](378.2KB)
Abstract:
We study topological cocycles of a class of non-isometric distal minimal homeomorphisms of multidimensional tori, introduced by Furstenberg in [5] as iterated skew product extensions by the torus, starting with an irrational rotation. We prove that there are no topological type ${III}_0$ cocycles of these homeomorphisms with values in an Abelian locally compact group. Moreover, under the assumption that the Abelian locally compact group has no non-trivial connected compact subgroup, we show that a topologically recurrent cocycle is always regular, i.e. it is topologically cohomologous to a cocycle with values only in the essential range. These properties are well-known for topological cocycles of minimal rotations on compact metric groups (cf. [6], [2], [9], and [10]), but the distal minimal homeomorphisms considered in this paper are far from the isometric behaviour of minimal rotations and do not admit rigidity times.
We study topological cocycles of a class of non-isometric distal minimal homeomorphisms of multidimensional tori, introduced by Furstenberg in [5] as iterated skew product extensions by the torus, starting with an irrational rotation. We prove that there are no topological type ${III}_0$ cocycles of these homeomorphisms with values in an Abelian locally compact group. Moreover, under the assumption that the Abelian locally compact group has no non-trivial connected compact subgroup, we show that a topologically recurrent cocycle is always regular, i.e. it is topologically cohomologous to a cocycle with values only in the essential range. These properties are well-known for topological cocycles of minimal rotations on compact metric groups (cf. [6], [2], [9], and [10]), but the distal minimal homeomorphisms considered in this paper are far from the isometric behaviour of minimal rotations and do not admit rigidity times.
2013, 33(9): 4323-4339
doi: 10.3934/dcds.2013.33.4323
+[Abstract](1758)
+[PDF](410.2KB)
Abstract:
In his foundational paper [20] , Mañé suggested that some aspects of the Oseledets splitting could be improved if one worked under $C^1$-generic conditions. He announced some powerful theorems, and suggested some lines to follow. Here we survey the state of the art and some recent advances in these directions.
In his foundational paper [20] , Mañé suggested that some aspects of the Oseledets splitting could be improved if one worked under $C^1$-generic conditions. He announced some powerful theorems, and suggested some lines to follow. Here we survey the state of the art and some recent advances in these directions.
2013, 33(9): 4341-4347
doi: 10.3934/dcds.2013.33.4341
+[Abstract](2271)
+[PDF](317.1KB)
Abstract:
For the 'infinite staircase' square tiled surface we classify the Radon invariant measures for the straight line flow, obtaining an analogue of the celebrated Veech dichotomy for an infinite genus lattice surface. The ergodic Radon measures arise from Lebesgue measure on a one parameter family of deformations of the surface. The staircase is a $\mathbb{Z}$-cover of the torus, reducing the question to the well-studied cylinder map.
For the 'infinite staircase' square tiled surface we classify the Radon invariant measures for the straight line flow, obtaining an analogue of the celebrated Veech dichotomy for an infinite genus lattice surface. The ergodic Radon measures arise from Lebesgue measure on a one parameter family of deformations of the surface. The staircase is a $\mathbb{Z}$-cover of the torus, reducing the question to the well-studied cylinder map.
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