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Discrete & Continuous Dynamical Systems - A
June 2006 , Volume 16 , Issue 2
A special Issue Dedicated to Anatole Katok On the Occasion of his 60th Birthday
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2006, 16(2): i-i
doi: 10.3934/dcds.2006.16.2i
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Abstract:
This issue of Discrete and Continuous Dynamical Systems is Anatole Katok at UC Berkeley dedicated to Anatole Katok and was conceived on the occasion of his 60th birthday. Anatole Katok was born in Washington, D.C. in 1944. In 1959 he placed second in the Moscow Mathematical Olympiad, and the year after entered Moscow State University, earning his mathematics doctorate in 1968 from Y. Sinai. After working in the department of mathematical methods at the Central Economics and Mathematics Institute for 10 years he emigrated with his family, moving via Vienna, Rome and Paris to the University ofMaryland. The position in the US allowed him to travel, attend and organize conferences, collaborate with other mathematicians and supervise students.
From this time on, he organized more conferences, special years and other events than anybody else in the dynamics community. During his five years at Maryland Katok was instrumental in the development of their dynamical systems school, and after moving to first Caltech and then Penn State he founded a strong group in dynamical systems at each of these institutions. The schools at Maryland and Penn State have become leading world centers. He has always been active in mentoring younger generations. During his student years he devoted much energy to mathematics olympiads and circles, at Penn State he has been the driving force behind the Mathematics Advanced Study Semesters program for especially strong mathematics undergraduates, and he has supervised more than two dozen doctoral students.
For more information please click the “Full Text” above.
This issue of Discrete and Continuous Dynamical Systems is Anatole Katok at UC Berkeley dedicated to Anatole Katok and was conceived on the occasion of his 60th birthday. Anatole Katok was born in Washington, D.C. in 1944. In 1959 he placed second in the Moscow Mathematical Olympiad, and the year after entered Moscow State University, earning his mathematics doctorate in 1968 from Y. Sinai. After working in the department of mathematical methods at the Central Economics and Mathematics Institute for 10 years he emigrated with his family, moving via Vienna, Rome and Paris to the University ofMaryland. The position in the US allowed him to travel, attend and organize conferences, collaborate with other mathematicians and supervise students.
From this time on, he organized more conferences, special years and other events than anybody else in the dynamics community. During his five years at Maryland Katok was instrumental in the development of their dynamical systems school, and after moving to first Caltech and then Penn State he founded a strong group in dynamical systems at each of these institutions. The schools at Maryland and Penn State have become leading world centers. He has always been active in mentoring younger generations. During his student years he devoted much energy to mathematics olympiads and circles, at Penn State he has been the driving force behind the Mathematics Advanced Study Semesters program for especially strong mathematics undergraduates, and he has supervised more than two dozen doctoral students.
For more information please click the “Full Text” above.
2006, 16(2): 279-305
doi: 10.3934/dcds.2006.16.279
+[Abstract](2424)
+[PDF](294.6KB)
Abstract:
The nonadditive thermodynamic formalism is a generalization of the classical thermodynamic formalism, in which the topological pressure of a single function $\phi$ is replaced by the topological pressure of a sequence of functions $\Phi=(\phi_n)_n$. The theory also includes a variational principle for the topological pressure, although with restrictive assumptions on $\Phi$. Our main objective is to provide a new class of sequences, the so-called almost additive sequences, for which it is possible not only to establish a variational principle, but also to discuss the existence and uniqueness of equilibrium and Gibbs measures. In addition, we give several characterizations of the invariant Gibbs measures, also in terms of an averaging procedure over the periodic points.
The nonadditive thermodynamic formalism is a generalization of the classical thermodynamic formalism, in which the topological pressure of a single function $\phi$ is replaced by the topological pressure of a sequence of functions $\Phi=(\phi_n)_n$. The theory also includes a variational principle for the topological pressure, although with restrictive assumptions on $\Phi$. Our main objective is to provide a new class of sequences, the so-called almost additive sequences, for which it is possible not only to establish a variational principle, but also to discuss the existence and uniqueness of equilibrium and Gibbs measures. In addition, we give several characterizations of the invariant Gibbs measures, also in terms of an averaging procedure over the periodic points.
2006, 16(2): 307-327
doi: 10.3934/dcds.2006.16.307
+[Abstract](1733)
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Abstract:
The existence of stable manifolds for nonuniformly hyperbolic trajectories is well know in the case of $C^{1+\alpha}$ dynamics, as proven by Pesin in the late 1970's. On the other hand, Pugh constructed a $C^1$ diffeomorphism that is not of class $C^{1+\alpha}$ for any $\alpha$ and for which there exists no stable manifold. The $C^{1+\alpha}$ hypothesis appears to be crucial in some parts of smooth ergodic theory, such as for the absolute continuity property and thus in the study of the ergodic properties of the dynamics. Nevertheless, we establish the existence of invariant stable manifolds for nonuniformly hyperbolic trajectories of a large family of maps of class at most $C^1$, by providing a condition which is weaker than the $C^{1+\alpha}$ hypothesis but which is still sufficient to establish a stable manifold theorem. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We note that our proof of the stable manifold theorem is new even in the case of $C^{1+\alpha}$ nonuniformly hyperbolic dynamics. In particular, the optimal $C^1$ smoothness of the invariant manifolds is obtained by constructing an invariant family of cones.
The existence of stable manifolds for nonuniformly hyperbolic trajectories is well know in the case of $C^{1+\alpha}$ dynamics, as proven by Pesin in the late 1970's. On the other hand, Pugh constructed a $C^1$ diffeomorphism that is not of class $C^{1+\alpha}$ for any $\alpha$ and for which there exists no stable manifold. The $C^{1+\alpha}$ hypothesis appears to be crucial in some parts of smooth ergodic theory, such as for the absolute continuity property and thus in the study of the ergodic properties of the dynamics. Nevertheless, we establish the existence of invariant stable manifolds for nonuniformly hyperbolic trajectories of a large family of maps of class at most $C^1$, by providing a condition which is weaker than the $C^{1+\alpha}$ hypothesis but which is still sufficient to establish a stable manifold theorem. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We note that our proof of the stable manifold theorem is new even in the case of $C^{1+\alpha}$ nonuniformly hyperbolic dynamics. In particular, the optimal $C^1$ smoothness of the invariant manifolds is obtained by constructing an invariant family of cones.
2006, 16(2): 329-341
doi: 10.3934/dcds.2006.16.329
+[Abstract](1629)
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Abstract:
Adapting techniques of Misiurewicz, for $1\leq r < \infty$ we give an explicit construction of $C^r$ maps with positive residual entropy. We also establish the behavior of symbolic extension entropy with respect to joinings, fiber products, products, powers and flows.
Adapting techniques of Misiurewicz, for $1\leq r < \infty$ we give an explicit construction of $C^r$ maps with positive residual entropy. We also establish the behavior of symbolic extension entropy with respect to joinings, fiber products, products, powers and flows.
2006, 16(2): 343-360
doi: 10.3934/dcds.2006.16.343
+[Abstract](1755)
+[PDF](274.8KB)
Abstract:
We call an ordered set $\mathbf{c} = (c(i): i \in \mathbb{N})$, of nonnegative extended real numbers $c(i)$, a universal skyscraper template if it is the distribution of first return times for every ergodic measure preserving transformation $T$ of an infinite Lebesgue measure space. If ∑ i$ c(i)<\infty$, we give a family of examples of ergodic infinite measure preserving transformations that do not admit c as a skyscraper template.
If the distribution $\mathbf{c}$ satisfies $\gcd\{i: c(i) >0 \} = 1 $, and if either of the conditions $c(I) = \infty$ (for some integer $I$), or $i n f_i \{c(i) \} > 0$ is satisfied, then $\mathbf{c}$ is a universal skyscraper template.
We call an ordered set $\mathbf{c} = (c(i): i \in \mathbb{N})$, of nonnegative extended real numbers $c(i)$, a universal skyscraper template if it is the distribution of first return times for every ergodic measure preserving transformation $T$ of an infinite Lebesgue measure space. If ∑ i$ c(i)<\infty$, we give a family of examples of ergodic infinite measure preserving transformations that do not admit c as a skyscraper template.
If the distribution $\mathbf{c}$ satisfies $\gcd\{i: c(i) >0 \} = 1 $, and if either of the conditions $c(I) = \infty$ (for some integer $I$), or $i n f_i \{c(i) \} > 0$ is satisfied, then $\mathbf{c}$ is a universal skyscraper template.
2006, 16(2): 361-365
doi: 10.3934/dcds.2006.16.361
+[Abstract](1938)
+[PDF](150.5KB)
Abstract:
Suppose $G$ is an infinite Abelian group that factorizes as the direct sum $G = A \oplus B$: i.e., the $B$-translates of the single tile $A$ evenly tile the group $G$ ($B$ is called the tile set). In this note, we consider conditions for another set $C \subset G$ to tile $G$ with the same tile set $B$. In an earlier paper, we answered a question of Sands regarding such tilings of $G$ when $A$ is a finite tile. We now consider extensions of Sands's question when $A$ is infinite. We offer two approaches to this question. The first approach involves a combinatorial condition used by Tijdeman and Sands. This condition completely characterizes when a set $C$ can tile $G$ with the tile set $B$; the condition is applied to simplify the proofs and extend some of Sands's results [8]. The second approach is measure theoretic and follows Eigen, Hajian, and Ito's work on exhaustive weakly wandering sets for ergodic infinite measure preserving transformations.
Suppose $G$ is an infinite Abelian group that factorizes as the direct sum $G = A \oplus B$: i.e., the $B$-translates of the single tile $A$ evenly tile the group $G$ ($B$ is called the tile set). In this note, we consider conditions for another set $C \subset G$ to tile $G$ with the same tile set $B$. In an earlier paper, we answered a question of Sands regarding such tilings of $G$ when $A$ is a finite tile. We now consider extensions of Sands's question when $A$ is infinite. We offer two approaches to this question. The first approach involves a combinatorial condition used by Tijdeman and Sands. This condition completely characterizes when a set $C$ can tile $G$ with the tile set $B$; the condition is applied to simplify the proofs and extend some of Sands's results [8]. The second approach is measure theoretic and follows Eigen, Hajian, and Ito's work on exhaustive weakly wandering sets for ergodic infinite measure preserving transformations.
2006, 16(2): 367-382
doi: 10.3934/dcds.2006.16.367
+[Abstract](1736)
+[PDF](259.8KB)
Abstract:
A configuration (i.e., a pair of points) in a Riemannian space $X$ is secure if all connecting geodesics can be blocked by a finite subset of $X$. A space is secure if all of its configurations are secure. Secure spaces seem to be rare.
If $X$ is an insecure space, it is natural to ask how big the set of insecure configurations is. We investigate this problem for flat surfaces, in particular for translation surfaces and polygons, from the viewpoint of measure theory.
Here is a sample of our results. Let $X$ be a lattice translation surface or a lattice polygon. Then the following dichotomy holds: i) The surface (polygon) $X$ is arithmetic. Then all configurations in $X$ are secure; ii) The surface (polygon) $X$ is nonarithmetic. Then almost all configurations in $X$ are insecure.
A configuration (i.e., a pair of points) in a Riemannian space $X$ is secure if all connecting geodesics can be blocked by a finite subset of $X$. A space is secure if all of its configurations are secure. Secure spaces seem to be rare.
If $X$ is an insecure space, it is natural to ask how big the set of insecure configurations is. We investigate this problem for flat surfaces, in particular for translation surfaces and polygons, from the viewpoint of measure theory.
Here is a sample of our results. Let $X$ be a lattice translation surface or a lattice polygon. Then the following dichotomy holds: i) The surface (polygon) $X$ is arithmetic. Then all configurations in $X$ are secure; ii) The surface (polygon) $X$ is nonarithmetic. Then almost all configurations in $X$ are insecure.
2006, 16(2): 383-392
doi: 10.3934/dcds.2006.16.383
+[Abstract](1911)
+[PDF](231.0KB)
Abstract:
Let $M_{\phi}$ denote the set of Borel probability measures invariant under a topological action $\phi$ on a compact metrizable space $X$. For a continuous function $f:X\to\R$, a measure $\mu\in\M_{\phi}$ is called $f$-maximizing if $\int f\, d\mu = s u p\{\int f dm:m\in\M_{\phi}\}$. It is shown that if $\mu$ is any ergodic measure in $\M_{\phi}$, then there exists a continuous function whose unique maximizing measure is $\mu$. More generally, if $\mathcal E$ is a non-empty collection of ergodic measures which is weak$^*$ closed as a subset of $\M_{\phi}$, then there exists a continuous function whose set of maximizing measures is precisely the closed convex hull of $\mathcal E$. If moreover $\phi$ has the property that its entropy map is upper semi-continuous, then there exists a continuous function whose set of equilibrium states is precisely the closed convex hull of $\mathcal E$.
Let $M_{\phi}$ denote the set of Borel probability measures invariant under a topological action $\phi$ on a compact metrizable space $X$. For a continuous function $f:X\to\R$, a measure $\mu\in\M_{\phi}$ is called $f$-maximizing if $\int f\, d\mu = s u p\{\int f dm:m\in\M_{\phi}\}$. It is shown that if $\mu$ is any ergodic measure in $\M_{\phi}$, then there exists a continuous function whose unique maximizing measure is $\mu$. More generally, if $\mathcal E$ is a non-empty collection of ergodic measures which is weak$^*$ closed as a subset of $\M_{\phi}$, then there exists a continuous function whose set of maximizing measures is precisely the closed convex hull of $\mathcal E$. If moreover $\phi$ has the property that its entropy map is upper semi-continuous, then there exists a continuous function whose set of equilibrium states is precisely the closed convex hull of $\mathcal E$.
2006, 16(2): 393-394
doi: 10.3934/dcds.2006.16.393
+[Abstract](1596)
+[PDF](133.1KB)
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We construct the simplest chaotic system with a two-point attractor on the plane.
We construct the simplest chaotic system with a two-point attractor on the plane.
2006, 16(2): 395-410
doi: 10.3934/dcds.2006.16.395
+[Abstract](1777)
+[PDF](276.0KB)
Abstract:
The topological entropy of piecewise affine maps is studied. It is shown that singularities may contribute to the entropy only if there is angular expansion and we bound the entropy via the expansion rates of the map. As a corollary, we deduce that non-expanding conformal piecewise affine maps have zero topological entropy. We estimate the entropy of piecewise affine skew-products. Examples of abnormal entropy growth are provided.
The topological entropy of piecewise affine maps is studied. It is shown that singularities may contribute to the entropy only if there is angular expansion and we bound the entropy via the expansion rates of the map. As a corollary, we deduce that non-expanding conformal piecewise affine maps have zero topological entropy. We estimate the entropy of piecewise affine skew-products. Examples of abnormal entropy growth are provided.
2006, 16(2): 411-433
doi: 10.3934/dcds.2006.16.411
+[Abstract](1873)
+[PDF](323.6KB)
Abstract:
We consider the horocycle flow associated to a $\Z^d$-cover of a compact hyperbolic surface. Such flows have no finite invariant measures, and infinitely many infinite ergodic invariant Radon measures. We prove that, up to normalization, only one of these infinite measures admits a generalized law of large numbers, and we identify such laws.
We consider the horocycle flow associated to a $\Z^d$-cover of a compact hyperbolic surface. Such flows have no finite invariant measures, and infinitely many infinite ergodic invariant Radon measures. We prove that, up to normalization, only one of these infinite measures admits a generalized law of large numbers, and we identify such laws.
2006, 16(2): 435-454
doi: 10.3934/dcds.2006.16.435
+[Abstract](2443)
+[PDF](258.6KB)
Abstract:
We study the nonadditive thermodynamic formalism for the class of almost-additive sequences of potentials. We define the topological pressure $P_Z(\Phi)$ of an almost-additive sequence $\Phi$, on a set $Z$. We give conditions which allow us to establish a variational principle for the topological pressure. We state conditions for the existence and uniqueness of equilibrium measures, and for subshifts of finite type the existence and uniqueness of Gibbs measures. Finally, we compare the results for almost-additive sequences to the thermodynamic formalism for the classical (additive) case [10] [11] [3], the sequences studied by Barreira [1], Falconer [5], and that of Feng and Lau [7], [6].
We study the nonadditive thermodynamic formalism for the class of almost-additive sequences of potentials. We define the topological pressure $P_Z(\Phi)$ of an almost-additive sequence $\Phi$, on a set $Z$. We give conditions which allow us to establish a variational principle for the topological pressure. We state conditions for the existence and uniqueness of equilibrium measures, and for subshifts of finite type the existence and uniqueness of Gibbs measures. Finally, we compare the results for almost-additive sequences to the thermodynamic formalism for the classical (additive) case [10] [11] [3], the sequences studied by Barreira [1], Falconer [5], and that of Feng and Lau [7], [6].
2006, 16(2): 455-461
doi: 10.3934/dcds.2006.16.455
+[Abstract](2102)
+[PDF](185.7KB)
Abstract:
We show that the iterated images of a Jacobian pair $f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ stabilize; that is, all the sets $f^k(\mathbb{C}^2)$ are equal for $k$ sufficiently large. More generally, let $X$ be a closed algebraic subset of $\mathbb{C}^N$, and let $f:X\rightarrow X$ be an open polynomial map with $X-f(X)$ a finite set. We show that the sets $f^k(X)$ stabilize, and for any cofinite subset $\Omega \subseteq X$ with $f(\Omega) \subseteq \Omega$, the sets $f^k(\Omega)$ stabilize. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.
We show that the iterated images of a Jacobian pair $f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ stabilize; that is, all the sets $f^k(\mathbb{C}^2)$ are equal for $k$ sufficiently large. More generally, let $X$ be a closed algebraic subset of $\mathbb{C}^N$, and let $f:X\rightarrow X$ be an open polynomial map with $X-f(X)$ a finite set. We show that the sets $f^k(X)$ stabilize, and for any cofinite subset $\Omega \subseteq X$ with $f(\Omega) \subseteq \Omega$, the sets $f^k(\Omega)$ stabilize. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.
2006, 16(2): 463-504
doi: 10.3934/dcds.2006.16.463
+[Abstract](2114)
+[PDF](401.8KB)
Abstract:
In the paper, we discuss two questions about degree $d$ smooth expanding circle maps, with $d \ge 2$. (i) We characterize the sequences of asymptotic length ratios which occur for systems with Hölder continuous derivative. The sequence of asymptotic length ratios are precisely those given by a positive Hölder continuous function $s$ (solenoid function) on the Cantor set $C$ of $d$-adic integers satisfying a functional equation called the matching condition. In the case of the $2$-adic integer Cantor set, the functional equation is
In the paper, we discuss two questions about degree $d$ smooth expanding circle maps, with $d \ge 2$. (i) We characterize the sequences of asymptotic length ratios which occur for systems with Hölder continuous derivative. The sequence of asymptotic length ratios are precisely those given by a positive Hölder continuous function $s$ (solenoid function) on the Cantor set $C$ of $d$-adic integers satisfying a functional equation called the matching condition. In the case of the $2$-adic integer Cantor set, the functional equation is
$ s (2x+1)= \frac{s (x)} {s (2x)}$ $1+\frac{1}{ s (2x-1)}-1. $
We also present a one-to-one correspondence between solenoid functions and affine classes of exponentially fast $d$-adic tilings of the real line that are fixed points of the $d$-amalgamation operator. (ii) We calculate the precise maximum possible level of smoothness for a representative of the system, up to diffeomorphic conjugacy, in terms of the functions $s$ and $cr(x)=(1+s(x))/(1+(s(x+1))^{-1})$. For example, in the Lipschitz structure on $C$ determined by $s$, the maximum smoothness is $C^{1+\alpha}$ for $0 < \alpha \le 1$ if and only if $s$ is $\alpha$-Hölder continuous. The maximum smoothness is $C^{2+\alpha}$ for $0 < \alpha \le 1$ if and only if $cr$ is $(1+\alpha)$-Hölder. A curious connection with Mostow type rigidity is provided by the fact that $s$ must be constant if it is $\alpha$-Hölder for $\alpha > 1$.
2006, 16(2): 505-512
doi: 10.3934/dcds.2006.16.505
+[Abstract](1892)
+[PDF](146.0KB)
Abstract:
We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.
We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.
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