116 results
Search Results
2. Measure-theoretic and topological entropy of operators on function spaces
- Author
-
TOMASZ DOWNAROWICZ and BARTOSZ FREJ
- Subjects
Pure mathematics ,Operator (computer programming) ,Markov chain ,Function space ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Axiomatic system ,Entropy (information theory) ,Topological entropy ,Uniqueness ,Mathematics ,Probability measure - Abstract
We study entropy of actions on function spaces with the focus on doubly stochastic operators on probability spaces and Markov operators on compact spaces. Using an axiomatic approach to entropy we prove that there is basically only one reasonable measure-theoretic entropy notion on doubly stochastic operators. By “reasonable” we mean extending the KolmogorovSinai entropy on measure-preserving transformations and satisfying some obvious continuity conditions for Hμ. In particular this establishes equality on such operators between the entropy notion introduced by R. Alicki, J. Andries, M. Fannes and P. Tuyls (a version of which was studied also by I.I. Makarov), another one introduced by E. Ghys, R. Langevin and P. Walczak, and our new definition introduced later in this paper. The key tool in proving this uniqueness is the discovery of a very general property of all doubly stochastic operators, which we call asymptotic lattice stability. Unlike the other explicit definitions of entropy mentioned above, ours satisfies many natural requirements already on the level of the function Hμ, and we prove that the limit defining hμ exists. The proof uses an integral representation of a stochastic operator obtained many years ago by A. Iwanik. In the topological part of the paper we introduce three natural definitions of topological entropy for Markov operators on C(X). Then we prove that all three are equal. Finally, we establish the partial variational principle: the topological entropy of a Markov operator majorizes the measure-theoretic entropy of this operator with respect to any of its invariant probability measures.
- Published
- 2005
3. On logarithmically small errors in the lattice point problem
- Author
-
M. M. Skriganov and A. N. Starkov
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics - Abstract
In the present paper we give an improvement of a previous result of the paper [M. M. Skriganov. Ergodic theory on $SL(n)$, diophantine approximations and anomalies in the lattice point problem. Inv. Math.132(1), (1998), 1–72, Theorem 2.2] on logarithmically small errors in the lattice point problem for polyhedra. This improvement is based on an analysis of hidden symmetries of the problem generated by the Weyl group for $SL(n,\mathbb{B})$. Let $UP$ denote a rotation of a given compact polyhedron $P\subset\mathbb{B}^n$ by an orthogonal matrix $U\in SO(n)$, $tUP$ a dilation of $UP$ by a parameter $t>0$ and $N(tUP)$ the number of integer points $\gamma\in\mathbb{Z}^n$ which fall into the polyhedron $tUP$. We show that for almost all rotations $U$ (in the sense of the Haar measure on $SO(n)$) the following asymptotic formula \[ N(t\UP)=t^n{\rm vol} P+ O((\log t)^{n-1+\varepsilon}),\quad t\to\infty, \] holds with arbitrarily small $\varepsilon>0$.
- Published
- 2000
4. Exponential mixing property for Hénon–Sibony maps of
- Author
-
Hao Wu
- Subjects
Property (philosophy) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mixing (physics) ,Exponential function ,Mathematics - Abstract
Let f be a Hénon–Sibony map, also known as a regular polynomial automorphism of $\mathbb {C}^k$ , and let $\mu $ be the equilibrium measure of f. In this paper we prove that $\mu $ is exponentially mixing for plurisubharmonic observables.
- Published
- 2021
5. Topological entropy of semi-dispersing billiards
- Author
-
S. Ferleger, Dmitri Burago, and A. Kononenko
- Subjects
Applied Mathematics ,General Mathematics ,Lorentz transformation ,Mathematical analysis ,Boundary (topology) ,Topological entropy ,Riemannian geometry ,Manifold ,Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,Flow (mathematics) ,symbols ,Sectional curvature ,Dynamical billiards ,Mathematics - Abstract
In this paper we continue to explore the applications of the connections between singular Riemannian geometry and billiard systems that were first used in [6] to prove estimates on the number of collisions in non-degenerate semi-dispersing billiards.In this paper we show that the topological entropy of a compact non-degenerate semi-dispersing billiard on any manifold of non-positive sectional curvature is finite. Also, we prove exponential estimates on the number of periodic points (for the first return map to the boundary of a simple-connected billiard table) and the number of periodic trajectories (for the billiard flow). In \S5 we prove some estimates for the topological entropy of Lorentz gas.
- Published
- 1998
6. Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies
- Author
-
Dongfeng Zhang and Junxiang Xu
- Subjects
Nonlinear system ,Matrix (mathematics) ,Class (set theory) ,Applied Mathematics ,General Mathematics ,Irrational number ,Mathematical analysis ,Zero (complex analysis) ,Perturbation (astronomy) ,Term (logic) ,Constant (mathematics) ,Mathematics - Abstract
In this paper we consider the following nonlinear quasi-periodic system:$$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $A$ is a $d\times d$ constant matrix of elliptic type, $\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$ is a small perturbation with $\unicode[STIX]{x1D716}$ as a small parameter, $h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$ as $x\rightarrow 0$, and $P,g$ and $h$ are all analytic quasi-periodic in $t$ with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$, where $\unicode[STIX]{x1D6FC}$ is irrational. It is proved that for most sufficiently small $\unicode[STIX]{x1D716}$, the system is reducible to the following form: $$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$ is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$, such that it goes to zero when $\unicode[STIX]{x1D716}$ does.
- Published
- 2020
7. Rigidity of higher-dimensional conformal Anosov systems
- Author
-
Rafael de la Llave
- Subjects
Rigidity (electromagnetism) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Jordan normal form ,Tangent ,Periodic orbits ,Conformal map ,Mathematics - Abstract
We show that Anosov systems in manifolds with trivial tangent bundles and with the property that the derivatives of the return maps at periodic orbits are multiples of the identity in the stable and unstable bundles are locally rigid. That is, any other smooth map, in a C 1 neighborhood such that it has the same Jordan normal form at corresponding periodic orbits is smoothly conjugate to it. This generalizes results of Castro and Moriyon (1997). We present several arguments for the main results. In particular, we use quasi- conformal regularity theory. We also extend the examples of an earlier paper of the author (1992) to show that some of the hypotheses we make in this paper are indeed necessary.
- Published
- 2002
8. Variational construction of positive entropy invariant measures of Lagrangian systems and Arnold diffusion
- Author
-
Siniša Slijepčević
- Subjects
Closed set ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fréchet derivative ,Dynamical Systems (math.DS) ,Topological entropy ,Invariant (physics) ,01 natural sciences ,Variational construction ,positive entropy ,Lagrangian systems ,Maxima and minima ,Variational method ,0103 physical sciences ,FOS: Mathematics ,Ergodic theory ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Arnold diffusion ,Primary 37J40, 37J45, Secondary: 37L45, 37L15, 34C28, 37A35, 37D25 ,Mathematics - Abstract
We develop a variational method for constructing positive entropy invariant measures of Lagrangian systems without assuming transversal intersections of stable and unstable manifolds, and without restrictions to the size of non-integrable perturbations. We apply it to a family of two and a half degrees of freedom a-priori unstable Lagrangians, and show that if we assume that there is no topological obstruction to diffusion (precisely formulated in terms of topological non-degeneracy of minima of the Peierl's barrier function), then there exists a vast family of "horsheshoes", such as "shadowing" ergodic positive entropy measures having precisely any closed set of invariant tori in its support. Furthermore, we give bounds on the topological entropy and the "drift acceleration" in any part of a region of instability in terms of a certain extremal value of the Fr\'{e}chet derivative of the action functional, generalizing the angle of splitting of separatrices. The method of construction is new, and relies on study of formally gradient dynamics of the action (coupled parabolic semilinear partial differential equations on unbounded domains). We apply recently developed techniques of precise control of the local evolution of energy (in this case the Lagrangian action), energy dissipation and flux. In Part II of the paper we will apply the theory to obtain sharp bounds for topological entropy and drift acceleration for the same class of equations in the case of small perturbations., Comment: Version 2: corrected typos
- Published
- 2018
9. Non-monotone periodic orbits of a rotational horseshoe
- Author
-
B. García and Valentín Mendoza
- Subjects
Forcing (recursion theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Annulus (mathematics) ,01 natural sciences ,0103 physical sciences ,Braid ,Periodic orbits ,Horseshoe orbit ,010307 mathematical physics ,0101 mathematics ,Non monotone ,Horseshoe (symbol) ,Mathematics - Abstract
In this paper, we present results for the forcing relation on the set of braid types of periodic orbits of a rotational horseshoe on the annulus. Precisely, we are concerned with a family of periodic orbits, called the Boyland family, and we prove that for each pair$(r,s)$of rational numbers with$rin$(0,1)$, there exists a non-monotone orbit$B_{r,s}$in this family which has pseudo-Anosov type and rotation interval$[r,s]$. Furthermore, the forcing relation among these orbits is given by the inclusion order on their rotation sets. It is also proved that the Markov partition associated to each Boyland orbit comes from a pruning map which projects to a bimodal circle map. This family also contains the Holmes orbits$H_{p/q}$, which are the largest for the forcing order among all the$(p,q)$-orbits of the rotational horseshoe.
- Published
- 2017
10. Phase transitions in long-range Ising models and an optimal condition for factors of -measures
- Author
-
Anders Johansson, Anders Öberg, and Mark Pollicott
- Subjects
Phase transition ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Inverse temperature ,01 natural sciences ,Range (mathematics) ,0103 physical sciences ,Ising model ,010307 mathematical physics ,Statistical physics ,0101 mathematics ,Mathematics ,Counterexample - Abstract
We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of $g$-measures. Indag. Math. (N.S.)22 (2011), 315–329] to a weaker condition, Berbee’s condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a $g$-measure with a continuous $g$-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.
- Published
- 2017
11. Poisson law for some non-uniformly hyperbolic dynamical systems with polynomial rate of mixing
- Author
-
Françoise Pène and Benoît Saussol
- Subjects
Normalization (statistics) ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Poisson distribution ,01 natural sciences ,law.invention ,010101 applied mathematics ,symbols.namesake ,Invertible matrix ,law ,symbols ,0101 mathematics ,Dynamical billiards ,Mathematics - Abstract
We consider some non-uniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs–Markov–Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball$B(x,r)$converges to a Poisson distribution as the radius$r\rightarrow 0$and after suitable normalization.
- Published
- 2015
12. Measurable rigidity for Kleinian groups
- Author
-
Woojin Jeon and Ken'ichi Ohshika
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,30F40, 37A40 ,Boundary (topology) ,Geometric Topology (math.GT) ,Rigidity (psychology) ,Dynamical Systems (math.DS) ,Type (model theory) ,01 natural sciences ,Divergence (computer science) ,Mathematics - Geometric Topology ,symbols.namesake ,Limit (category theory) ,0103 physical sciences ,FOS: Mathematics ,symbols ,Equivariant map ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics ,Möbius transformation - Abstract
Let $G, H$ be two Kleinian groups with homeomorphic quotients $\mathbb H^3/G$ and $\mathbb H^3/H$. We assume that $G$ is of divergence type, and consider the Patterson-Sullivan measures of $G$ and $H$. The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map $\widehat k$ from the limit set $\Lambda_G$ of $G$ to that of $H$ is either the restriction of a M\"{o}bius transformation or totally singular. In this paper, we shall show that such $\widehat k$ always exists. In fact, we shall construct $\widehat k$ concretely from the Cannon-Thurston maps of $G$ and $H$., Comment: 16 pages, no figures
- Published
- 2015
13. On the ergodicity of hyperbolic Sinaĭ–Ruelle–Bowen measures: the constant unstable dimension case
- Author
-
Michihiro Hirayama and Naoya Sumi
- Subjects
Surface (mathematics) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Ergodicity ,Hyperbolic manifold ,Context (language use) ,01 natural sciences ,Manifold ,010101 applied mathematics ,Almost everywhere ,Uniqueness ,0101 mathematics ,Mathematics ,Probability measure - Abstract
In this paper we consider diffeomorphisms preserving hyperbolic Sinaĭ–Ruelle–Bowen (SRB) probability measures${\it\mu}$having intersections for almost every pair of the stable and unstable manifolds. In this context, when the dimension of the unstable manifold is constant almost everywhere, we show the ergodicity of${\it\mu}$. As an application we obtain another proof of the ergodicity of a hyperbolic SRB measure for transitive surface diffeomorphisms, which is shown by Rodriguez Hertz, Rodriguez Hertz, Tahzibi and Ures [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces.Comm. Math. Phys.306(1) (2011), 35–49].
- Published
- 2015
14. On the reducibility of two-dimensional linear quasi-periodic systems with small parameter
- Author
-
Xuezhu Lu and Junxiang Xu
- Subjects
Matrix (mathematics) ,Lebesgue measure ,Differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Sense (electronics) ,Quasi periodic ,Constant (mathematics) ,Coefficient matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we consider a linear real analytic quasi-periodic system of two differential equations, whose coefficient matrix analytically depends on a small parameter and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption of the small parameter, we prove that the system is reducible for most of the sufficiently small parameters in the sense of the Lebesgue measure.
- Published
- 2014
15. A dynamical-geometric characterization of the geodesic flows of negatively curved locally symmetric spaces
- Author
-
Yong Fang
- Subjects
Geodesic ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Characterization (mathematics) ,Mathematics - Abstract
In this paper we prove the following rigidity result: let ${\it\varphi}$ be a $C^{\infty }$ topologically mixing transversely symplectic Anosov flow. If (i) its weak stable and weak unstable distributions are $C^{\infty }$ and (ii) its Hamenstädt metrics are sub-Riemannian, then up to finite covers and a constant change of time scale, ${\it\varphi}$ is $C^{\infty }$ flow conjugate to the geodesic flow of a closed locally symmetric Riemannian space of rank one.
- Published
- 2014
16. Variations on a central limit theorem in infinite ergodic theory
- Author
-
Damien Thomine
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Ergodic theory ,Infinite product ,Ergodic hypothesis ,Invariant measure ,Ergodic Ramsey theory ,Stationary ergodic process ,Dynamical system ,Empirical process ,Mathematics - Abstract
In a previous paper the author proved a distributional convergence for the Birkhoff sums of functions of null average defined over a dynamical system with an infinite, invariant, ergodic measure, akin to a central limit theorem. Here we extend this result to larger classes of observables, with milder smoothness conditions, and to larger classes of dynamical systems, which may no longer be mixing. A special emphasis is given to continuous time systems: semi-flows, flows, and $\mathbb{Z}^{d}$-extensions of flows. The latter generalization is applied to the geodesic flow on $\mathbb{Z}^{d}$-periodic manifolds of negative sectional curvature.
- Published
- 2014
17. A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model
- Author
-
Hayato Chiba
- Subjects
Spectral theory ,Applied Mathematics ,General Mathematics ,Kuramoto model ,Mathematical analysis ,Hilbert space ,Dynamical Systems (math.DS) ,Rigged Hilbert space ,Space (mathematics) ,Bifurcation diagram ,symbols.namesake ,Bifurcation theory ,FOS: Mathematics ,symbols ,Mathematics - Dynamical Systems ,Center manifold ,Mathematics - Abstract
The Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as a coupled phase oscillators. In this paper, a bifurcation structure of the infinite dimensional Kuramoto model is investigated. For a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, the spectral theory on a space of generalized functions is developed with the aid of a rigged Hilbert space to avoid a continuous spectrum on the imaginary axis. Although the linear operator has an unbounded continuous spectrum on a Hilbert space, it is shown that it admits a spectral decomposition consisting of a countable number of eigenfunctions on a space of generalized functions. The semigroup generated by the linear operator is calculated by using the spectral decomposition to prove the linear stability of a steady state of the system. The center manifold theory is also developed on a space of generalized functions. It is proved that there exists a finite dimensional center manifold on a space of generalized functions, while a center manifold on a Hilbert space is of infinite dimensional because of the continuous spectrum on the imaginary axis. The results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto. If the coupling strength $K$ between oscillators is smaller than some threshold $K_c$, the de-synchronous state proves to be asymptotically stable, and if $K$ exceeds $K_c$, a nontrivial solution, which corresponds to the synchronization, bifurcates from the de-synchronous state., Comment: It will be published in Ergodic Theory and Dynamical Systems
- Published
- 2013
18. Continuation and bifurcation associated to the dynamical spectral sequence
- Author
-
Robert D. Franzosa, K. A. de Rezende, and M. R. Da Silveira
- Subjects
Continuation ,Pure mathematics ,Sequence ,Flow (mathematics) ,Applied Mathematics ,General Mathematics ,Spectral sequence ,Mathematical analysis ,Saddle-node bifurcation ,Bifurcation diagram ,Bifurcation ,Connection (mathematics) ,Mathematics - Abstract
In this paper we consider a filtered chain complex $C$ and its differential given by a connection matrix $\Delta $ which determines an associated spectral sequence $({E}^{r} , {d}^{r} )$. We present an algorithm which sweeps the connection matrix in order to span the modules ${E}^{r} $ in terms of bases of $C$ and gives the differentials ${d}^{r} $. In this process a sequence of similar connection matrices and associated transition matrices are produced. This algebraic procedure can be viewed as a continuation, where the transition matrices give information about the bifurcation behavior. We introduce directed graphs, called flow and bifurcation schematics, that depict bifurcations that could occur if the sequence of connection matrices and transition matrices were realized in a continuation of a Morse decomposition, and we present a dynamic interpretation theorem that provides conditions on a parameterized family of flows under which such a continuation could occur.
- Published
- 2013
19. Analytic moduli for unfoldings of germs of generic analytic diffeomorphisms with a codimension parabolic point
- Author
-
Christiane Rousseau
- Subjects
Applied Mathematics ,General Mathematics ,Global analytic function ,Mathematical analysis ,Point (geometry) ,Codimension ,Moduli ,Mathematics - Abstract
In this paper we provide a complete modulus of analytic classification for germs of generic analytic families of diffeomorphisms which unfold a parabolic fixed point of codimension$k$. We start by showing that a generic family can be ‘prepared’, i.e. brought to a prenormal form${f}_{\epsilon } (z)$in which the multi-parameter$\epsilon $is almost canonical (up to an action of$ \mathbb{Z} / k \mathbb{Z} $). As in the codimension one case treated in P. Mardešić, R. Roussarie and C. Rousseau [Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms.Mosc. Math. J. 4(2004), 455–498], we show that the Ecalle–Voronin modulus can be unfolded to give a complete modulus for such germs. For this purpose, we define unfolded sectors in$z$-space that constitute natural domains on which the map${f}_{\epsilon } $can be brought to normal form in an almost unique way. The comparison of these normalizing changes of coordinates on the different sectors forms the analytic part of the modulus. This construction is performed on sectors in the multi-parameter space$\epsilon $such that the closure of their union provides a neighborhood of the origin in parameter space.
- Published
- 2013
20. Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schrödinger cocycle with weak Liouville frequency
- Author
-
Jiangong You and Shiwen Zhang
- Subjects
Subharmonic ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Hölder condition ,Lyapunov exponent ,Schrödinger equation ,symbols.namesake ,Quasiperiodic function ,symbols ,Density of states ,Base frequency ,Schrödinger's cat ,Mathematics - Abstract
For analytic quasiperiodic Schrödinger cocycles, Goldshtein and Schlag [Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions.Ann. of Math.(2)154(2001), 155–203] proved that the Lyapunov exponent is Hölder continuous provided that the base frequency$\omega $satisfies a strong Diophantine condition. In this paper, we give a refined large deviation theorem, which implies the Hölder continuity of the Lyapunov exponent for all Diophantine frequencies$\omega $, even for weak Liouville$\omega $, which improves the result of Goldshtein and Schlag.
- Published
- 2013
21. Viscous stability of quasi-periodic tori
- Author
-
Yingfei Yi, Jun Yan, and Zhenguo Liang
- Subjects
Holder exponent ,Applied Mathematics ,General Mathematics ,Diophantine equation ,Mathematical analysis ,Hölder condition ,Torus ,Lipschitz continuity ,Viscosity ,symbols.namesake ,symbols ,Quasi periodic ,Hamiltonian (quantum mechanics) ,Mathematical physics ,Mathematics - Abstract
This paper is devoted to the study of$P$-regularity of viscosity solutions$u(x,P)$,$P\in {\Bbb R}^n$, of a smooth Tonelli Lagrangian$L:T {\Bbb T}^n \rightarrow {\Bbb R}$characterized by the cell equation$H(x,P+D_xu(x,P))=\overline {H}(P)$, where$H: T^* {\Bbb T}^n\rightarrow {\Bbb R}$denotes the Hamiltonian associated with$L$and$\overline {H}$is the effective Hamiltonian. We show that if$P_0$corresponds to a quasi-periodic invariant torus with a non-resonant frequency, then$D_xu(x,P)$is uniformly Hölder continuous in$P$at$P_0$with Hölder exponent arbitrarily close to$1$, and if both$H$and the torus are real analytic and the frequency vector of the torus is Diophantine, then$D_xu(x,P)$is uniformly Lipschitz continuous in$P$at$P_0$, i.e., there is a constant$C\gt 0$such that$\|D_xu(\cdot ,P)-D_xu(\cdot ,P_0)\|_{\infty }\le C\|P-P_0\|$for$\|P-P_0\|\ll 1$. Similar P-regularity of the Peierls barriers associated with$L(x,v)- \langle P,v \rangle $is also obtained.
- Published
- 2012
22. Existence of generic cubic homoclinic tangencies for Hénon maps
- Author
-
Shin Kiriki and Teruhiko Soma
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Tangent ,Heteroclinic cycle ,symbols.namesake ,Poincaré conjecture ,Attractor ,Dissipative system ,symbols ,Order (group theory) ,Homoclinic orbit ,Diffeomorphism ,Mathematics - Abstract
In this paper, we show that the Hénon map $\varphi _{a,b}$ has a generically unfolding cubic tangency for some $(a,b)$ arbitrarily close to $(-2,0)$ by applying results of Gonchenko, Shilnikov and Turaev [On models with non-rough Poincaré homoclinic curves. Physica D 62(1–4) (1993), 1–14; Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits. Chaos 6(1) (1996), 15–31; On Newhouse domains of two-dimensional diffeomorphisms which are close to a diffeomorphism with a structurally unstable heteroclinic cycle. Proc. Steklov Inst. Math.216 (1997), 70–118; Homoclinic tangencies of an arbitrary order in Newhouse domains. Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. 67 (1999), 69–128, translation in J. Math. Sci. 105 (2001), 1738–1778; Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity 20 (2007), 241–275]. Combining this fact with theorems in Kiriki and Soma [Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies. Nonlinearity 21(5) (2008), 1105–1140], one can observe the new phenomena in the Hénon family, appearance of persistent antimonotonic tangencies and cubic polynomial-like strange attractors.
- Published
- 2012
23. Family of piecewise expanding maps having singular measure as a limit of ACIMs
- Author
-
Abraham Boyarsky, Zhenyang Li, Harald Proppe, Paweł Góra, and Peyman Eslami
- Subjects
Piecewise linear function ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Piecewise ,Singular measure ,Limit (mathematics) ,Invariant measure ,Singular point of a curve ,Invariant (mathematics) ,Measure (mathematics) ,Mathematics - Abstract
Keller [Stochastic stability in some chaotic dynamical systems. Monatsh. Math.94(4) (1982), 313–333] introduced families of W-shaped maps that can have a great variety of behaviors. As a family approaches a limit W map, he observed behavior that was either described by a probability density function (PDF) or by a singular point measure. Based on this, Keller conjectured that instability of the absolutely continuous invariant measure (ACIM) can result only from the existence of small invariant neighborhoods of the fixed critical point of the limit map. In this paper, we show that the conjecture is not true. We construct a very simple family of W-maps with ACIMs supported on the whole interval, whose limiting dynamical behavior is captured by a singular measure. Key to the analysis is the use of a general formula for invariant densities of piecewise linear and expanding maps [P. Góra. Invariant densities for piecewise linear maps of interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583].
- Published
- 2011
24. Piecewise isometries, uniform distribution and 3log 2−π2/8
- Author
-
Yitwah Cheung, Anthony Quas, and Arek Goetz
- Subjects
Uniform distribution (continuous) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Piecewise ,Mathematics - Abstract
We use analytic tools to study a simple family of piecewise isometries of the plane parameterized by an angle. In previous work, we showed the existence of large numbers of periodic points, each surrounded by a ‘periodic island’. We also proved conservativity of the systems as infinite measure-preserving transformations. In experiments it is observed that the periodic islands fill up a large part of the phase space and it has been asked whether the periodic islands form a set of full measure. In this paper we study the periodic islands around an important family of periodic orbits and demonstrate that for all angle parameters that are irrational multiples of π, the islands have asymptotic density in the plane of 3log 2−π2/8≈0.846.
- Published
- 2011
25. Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform
- Author
-
Masato Tsujii
- Subjects
symbols.namesake ,Transfer (group theory) ,Fourier transform ,Function space ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Spectral properties ,symbols ,Mathematics - Abstract
This paper is about spectral properties of transfer operators for contact Anosov flows. The main result gives the essential spectral radii of the transfer operators acting on an appropriate function space exactly and improves the previous result in Tsujii [Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity23 (2010), 1495–1545]. Also, we provide a simplified proof by using the so-called Fourier–Bros–Iagolnitzer (FBI) (or Bargmann) transform.
- Published
- 2011
26. Tail pressure and the tail entropy function
- Author
-
Yuan Li, Wen-Chiao Cheng, and Ercai Chen
- Subjects
Binary entropy function ,Topological pressure ,Entropy (classical thermodynamics) ,Variational principle ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Calculus ,Direct proof ,Extension (predicate logic) ,Invariant (mathematics) ,Mathematics - Abstract
Burguet [A direct proof of the tail variational principle and its extension to maps. Ergod. Th. & Dynam. Sys.29 (2009), 357–369] presented a direct proof of the variational principle of tail entropy and extended Downarowicz’s results to a non-invertible case. This paper defines and discusses tail pressure, which is an extension of tail entropy for continuous transformations. This study reveals analogs of many known results of topological pressure. Specifically, a variational principle is provided and some applications of tail pressure, such as the investigation of invariant measures and equilibrium states, are also obtained.
- Published
- 2011
27. Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation
- Author
-
Junxiang Xu and Shunjun Jiang
- Subjects
Equilibrium point ,Nonlinear system ,Differential equation ,Applied Mathematics ,General Mathematics ,Degenerate energy levels ,Mathematical analysis ,Perturbation (astronomy) ,Quasi periodic ,Mathematics - Abstract
In this paper, using the Kolmogorov–Arnold–Moser method we prove reducibility of a class of nonlinear quasi-periodic differential equation with degenerate equilibrium point under small perturbation and obtain a quasi-periodic solution near the equilibrium point.
- Published
- 2010
28. Spectral sequences in Conley’s theory
- Author
-
Octavian Cornea, K. A. de Rezende, and M. R. Da Silveira
- Subjects
Path (topology) ,Pure mathematics ,Basis (linear algebra) ,Flow (mathematics) ,Applied Mathematics ,General Mathematics ,Spectral sequence ,Mathematical analysis ,Filtration (mathematics) ,Gravitational singularity ,Connection (algebraic framework) ,Type (model theory) ,Mathematics - Abstract
In this paper, we analyse the dynamics encoded in the spectral sequence (Er,dr) associated with certain Conley theory connection maps in the presence of an ‘action’ type filtration. More specifically, we present an algorithm for finding a chain complex C and its differential; the method uses a connection matrix Δ to provide a system that spans Er in terms of the original basis of C and to identify all of the differentials drp:Erp→Erp−r. In exploring the dynamical implications of a non-zero differential, we prove the existence of a path that joins the singularities generating E0p and E0p−r in the case where a direct connection by a flow line does not exist. This path is made up of juxtaposed orbits of the flow and of the reverse flow, and proves to be important in some applications.
- Published
- 2009
29. Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles
- Author
-
M. S. Schulteis, Svetlana Jitomirskaya, and D. A. Koslover
- Subjects
Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Quantum dynamics ,Quasiperiodic function ,Orthogonal polynomials on the unit circle ,Mathematical analysis ,symbols ,Lyapunov equation ,Lyapunov exponent ,Space (mathematics) ,Mathematics - Abstract
It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasiperiodic cocycles. In this paper we show that it is continuous in the analytic category. Our corollaries include continuity of the Lyapunov exponent associated with general quasiperiodic Jacobi matrices or orthogonal polynomials on the unit circle, in various parameters, and applications to the study of quantum dynamics.
- Published
- 2009
30. Rigidity of the Weyl chamber flow, and vanishing theorems of Matsushima and Weil
- Author
-
Masahico Kanai
- Subjects
Transverse plane ,Pure mathematics ,Rigidity (electromagnetism) ,Applied Mathematics ,General Mathematics ,Infinitesimal ,Mathematical analysis ,Lie group ,Mathematics - Abstract
The aim of the present paper is to reveal an unforeseen link between the classical vanishing theorems of Matsushima and Weil, on the one hand, and rigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank non-compact Lie group, on the other. The connection is established via ‘transverse extension theorems’: roughly speaking, they claim that a tangential 1-form of the orbit foliation of the Weyl chamber flow that is tangentially closed (and satisfies a certain mild additional condition) can be extended to a closed 1-form on the whole space in a canonical manner. In particular, infinitesimal rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.
- Published
- 2009
31. Embedding diffeomorphisms in flows in Banach spaces
- Author
-
Xiang Zhang
- Subjects
Floquet theory ,Pure mathematics ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Banach space ,Local diffeomorphism ,Context (language use) ,Flow (mathematics) ,Embedding ,Vector field ,Hyperbolic equilibrium point ,Mathematics - Abstract
This paper concerns the problem of embedding, in the flow of an autonomous vector field, a local diffeomorphism near a hyperbolic fixed point in a Banach space. To solve the problem, we first extend Floquet theory to Banach spaces, and then prove that two C∞ hyperbolic diffeomorphisms are formally equivalent if and only if they are C∞-equivalent. The latter result is a version, in the Banach space context, of a classical theorem by Chen.
- Published
- 2009
32. Natural invariant measures, divergence points and dimension in one-dimensional holomorphic dynamics
- Author
-
William Nathan Ingle, Christian Wolf, and Jacie L. Kaufmann
- Subjects
Pure mathematics ,Dense set ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Holomorphic function ,Riemann sphere ,Identity theorem ,symbols.namesake ,Plurisubharmonic function ,Hausdorff dimension ,symbols ,Invariant measure ,Invariant (mathematics) ,Mathematics - Abstract
In this paper we discuss the dimension-theoretical properties of rational maps on the Riemann sphere. In particular, we study the existence and uniqueness of generalized physical measures for several classes of maps including hyperbolic, parabolic, non-recurrent and topological Collet–Eckmann maps. These measures have the property that their typical points have maximal Hausdorff dimension. On the other hand, we prove that the set of divergence points (the set of points which are non-typical for any invariant measure) also has maximal Hausdorff dimension. Finally, we prove that if (fa)ais a holomorphic family of stable rational maps, then the dimensiond(fa) is a continuous and plurisubharmonic function of the parametera. In particular,d(f) varies continuously and plurisubharmonically on an open and dense subset ofRatd, the space of all rational maps with degreed≥2.
- Published
- 2009
33. Morse and Lyapunov spectra and dynamics on flag bundles
- Author
-
Luiz A. B. San Martin and Lucas Seco
- Subjects
Weyl group ,Iwasawa decomposition ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Vector bundle ,Frame bundle ,Principal bundle ,symbols.namesake ,Flow (mathematics) ,Associated bundle ,symbols ,Mathematics ,Flag (geometry) - Abstract
In this paper we study characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group G. Projection against the Iwasawa decomposition G=KAN defines an additive cocycle over the flow with values in 𝔞=log A. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. A symmetric property of these spectral sets is proved, namely invariance under the Weyl group. We also prove that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered.
- Published
- 2009
34. Global attractors of analytic plane flows
- Author
-
Víctor Jiménez López and Daniel Peralta-Salas
- Subjects
Polynomial ,Dynamical systems theory ,Plane (geometry) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Attractor ,Differentiable function ,Characterization (mathematics) ,Mathematics - Abstract
In this paper the global attractors of analytic and polynomial plane flows are characterized up to homeomorphisms. Following on from previous results for continuous and differentiable dynamical systems, our theorem completes the characterization of the global attractors of plane flows.
- Published
- 2009
35. Multifractal analysis of the Lyapunov exponent for the backward continued fraction map
- Author
-
Godofredo Iommi
- Subjects
Markov chain ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Lyapunov exponent ,Multifractal system ,Fixed point ,symbols.namesake ,Inflection point ,symbols ,Countable set ,Uniqueness ,Pressure function ,Mathematics - Abstract
In this paper we study the multifractal spectrum of Lyapunov exponents for interval maps with infinitely many branches and a parabolic fixed point. It turns out that, in strong contrast with the hyperbolic case, the domain of the spectrum is unbounded and points of non-differentiability might exist. Moreover, the spectrum is not concave. We establish conditions that ensure the existence of inflection points. To the best of our knowledge this is the first time that conditions of this type have been given. We also study the thermodynamic formalism for such maps. We prove that the pressure function is real analytic in a certain interval and then becomes equal to zero. We also discuss the existence and uniqueness of equilibrium measures. In order to do so, we introduce a family of countable Markov shifts that can be thought of as a generalization of the renewal shift.
- Published
- 2009
36. On the critical dimensions of product odometers
- Author
-
Anthony H. Dooley and Genevieve Mortiss
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Entropy (information theory) ,Critical dimension ,Odometer ,Covering lemma ,Mathematics - Abstract
Mortiss introduced the notion of critical dimension of a non-singular action, a measure of the order of growth of sums of Radon derivatives. The critical dimension was shown to be an invariant of metric isomorphism; this invariant was calculated for two-point product odometers and shown to coincide, in certain cases, with the average coordinate entropy. In this paper we extend the theory to apply to all product odometers, introduce upper and lower critical dimensions, and prove a Katok-type covering lemma.
- Published
- 2009
37. Chaotic period doubling
- Author
-
Marco Martens, V. V. M. S. Chandramouli, Charles Tresser, and W. de Melo
- Subjects
PROOF ,Mathematics::Dynamical Systems ,RENORMALIZATION ,General Mathematics ,Dynamical Systems (math.DS) ,Topological entropy ,Fixed point ,CIRCLE ,Renormalization ,FEIGENBAUM CONJECTURES ,Operator (computer programming) ,Attractor ,FOS: Mathematics ,DIFFEOMORPHISMS ,Uniqueness ,Mathematics - Dynamical Systems ,Mathematics ,Period-doubling bifurcation ,Computer Science::Information Retrieval ,Applied Mathematics ,Mathematical analysis ,Universality (dynamical systems) ,CONJUGATION ,UNIVERSALITY ,POINTS ,NON-LINEAR TRANSFORMATIONS - Abstract
The period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space ofC2+αunimodal maps, forα>0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space ofC2unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to geta prioribounds. In this smoother class, calledC2+∣⋅∣, the failure of hyperbolicity is tamer than inC2. Things get much worse with just a bit less smoothness thanC2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space ofC1+Lipunimodal maps has infinite topological entropy.
- Published
- 2009
38. Normal forms for almost periodic differential systems
- Author
-
Hao Wu, Jaume Llibre, and Weigu Li
- Subjects
symbols.namesake ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,symbols ,Lyapunov exponent ,Type (model theory) ,Differential systems ,Conjugate ,Mathematics - Abstract
In this paper we prove smooth conjugate theorems of Sternberg type for almost periodic differential systems, based on the Lyapunov exponents of the corresponding reduced systems.
- Published
- 2009
39. Theorem of Sternberg–Chen modulo the central manifold for Banach spaces
- Author
-
Victoria Rayskin
- Subjects
Pure mathematics ,Functional analysis ,biology ,Applied Mathematics ,General Mathematics ,Modulo ,Eberlein–Šmulian theorem ,Mathematical analysis ,Banach space ,Banach manifold ,biology.organism_classification ,Manifold ,Chen ,Lp space ,Mathematics - Abstract
We consider C∞-diffeomorphisms on a Banach space with a fixed point 0 and linear part L. Suppose that these diffeomorphisms have C∞ non-contracting and non-expanding invariant manifolds, and formally conjugate along their intersection (the center). We prove that they admit local C∞ conjugation. In particular, subject to non-resonance conditions, there exists a local C∞ linearization of the diffeomorphisms. It also follows that a family of germs with a hyperbolic linear part admits a C∞ linearization, which has C∞ dependence on the parameter of the linearizing family. The results are proved under the assumption that the Banach space allows a special extension of the maps. We discuss corresponding properties of Banach spaces. The proofs of this paper are based on the technique, developed in the works of Belitskii [Funct. Anal. Appl.18 (1984), 238–239; Funct. Anal. Appl.8 (1974), 338–339].
- Published
- 2009
40. Lower semicontinuity of attractors for non-autonomous dynamical systems
- Author
-
Alexandre N. Carvalho, James C. Robinson, and José A. Langa
- Subjects
Mathematics::Dynamical Systems ,Dynamical systems theory ,Linearization ,Differential equation ,Applied Mathematics ,General Mathematics ,Exponential dichotomy ,Mathematical analysis ,Attractor ,Banach space ,QA ,Mathematics::Geometric Topology ,Mathematics - Abstract
This paper is concerned with the lower semicontinuity of attractors for semilinear non-autonomous differential equations in Banach spaces. We require the unperturbed attractor to be given as the union of unstable manifolds of time-dependent hyperbolic solutions, generalizing previous results valid only for gradient-like systems in which the hyperbolic solutions are equilibria. The tools employed are a study of the continuity of the local unstable manifolds of the hyperbolic solutions and results on the continuity of the exponential dichotomy of the linearization around each of these solutions.
- Published
- 2009
41. Deformation of Brody curves and mean dimension
- Author
-
Masaki Tsukamoto
- Subjects
Mathematics - Differential Geometry ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Deformation theory ,Holomorphic function ,32H30 ,Parameter space ,Deformation (meteorology) ,Space (mathematics) ,Quantitative Biology::Other ,Differential Geometry (math.DG) ,Dimension (vector space) ,FOS: Mathematics ,Projective space ,Complex Variables (math.CV) ,Complex plane ,Mathematics - Abstract
The main purpose of this paper is to show that ideas of deformation theory can be applied to "infinite dimensional geometry". We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves., Comment: 18 pages
- Published
- 2009
42. Dynamics of two-dimensional Blaschke products
- Author
-
Michael Shub and Enrique R. Pujals
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Blaschke product ,Mathematical analysis ,Dynamics (mechanics) ,Open set ,Composition (combinatorics) ,Measure (mathematics) ,symbols.namesake ,Attractor ,symbols ,Diffeomorphism ,Mathematics - Abstract
In this paper we study the dynamics on $\mathbb {T}^2$ and $\mathbb {C}^2$ of a two-dimensional Blaschke product. We prove that in the case when the Blaschke product is a diffeomorphism of $\mathbb {T}^2$ with all periodic points hyperbolic then the dynamics is hyperbolic. If a two-dimensional Blaschke product diffeomorphism of $\mathbb {T}^2$ is embedded in a two-dimensional family given by composition with translations of $\mathbb {T}^2$, then we show that there is a non-empty open set of parameter values for which the dynamics is Anosov or has an expanding attractor with a unique SRB measure.
- Published
- 2008
43. Cantor sets of circles of Sierpiński curve Julia sets
- Author
-
Robert L. Devaney
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Open set ,Subshift of finite type ,Julia set ,Cantor set ,Combinatorics ,symbols.namesake ,Unit circle ,Connectedness locus ,symbols ,Sierpiński curve ,Invariant (mathematics) ,Mathematics - Abstract
Our goal in this paper is to give an example of a one-parameter family of rational maps for which, in the parameter plane, there is a Cantor set of simple closed curves consisting of parameters for which the corresponding Julia set is a Sierpiński curve. Hence, the Julia sets for each of these parameters are homeomorphic. However, each of the maps in this set is dynamically distinct from (i.e. not topologically conjugate to) any other map in this set (with only finitely many exceptions). We also show that, in the dynamical plane for any map drawn from a large open set in the connectedness locus in this family, there is a Cantor set of invariant simple closed curves on which the map is conjugate to the product of certain subshifts of finite type with the maps $z \mapsto \pm z^n$ on the unit circle.
- Published
- 2007
44. Infinite ergodic theory for Kleinian groups
- Author
-
Manuel Stadlbauer and Bernd O. Stratmann
- Subjects
Mathematics::Dynamical Systems ,Kleinian group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Hyperbolic manifold ,16. Peace & justice ,01 natural sciences ,Corollary ,Number theory ,0103 physical sciences ,Exponent ,Ergodic theory ,Markov property ,010307 mathematical physics ,Invariant measure ,0101 mathematics ,Mathematics - Abstract
In this paper we use infinite ergodic theory to study limit sets of essentially free Kleinian groups which may have parabolic elements of arbitrary rank. By adapting a method of Adler, we construct a section map S for the geodesic flow on the associated hyperbolic manifold. We then show that this map has the Markov property and that it is conservative and ergodic with respect to the invariant measure induced by the Liouville–Patterson measure. Furthermore, we obtain that S is rationally ergodic with respect to different types of return sequences (an), which are governed by the exponent of convergence . We then give applications to number theory and to the statistics of cuspidal windings. Also, as a corollary we obtain a special case of Sullivan's result that the geodesic flow on a geometrically finite hyperbolic manifold is ergodic with respect to the Liouville–Patterson measure.
- Published
- 2005
45. Sierpinski-curve Julia sets and singular perturbations of complex polynomials
- Author
-
Paul Blanchard, Pradipta Seal, Robert L. Devaney, Daniel M. Look, and Yakov Shapiro
- Subjects
Singular perturbation ,Pure mathematics ,Mathematics::Dynamical Systems ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Open set ,Julia set ,Sierpinski triangle ,Filled Julia set ,symbols.namesake ,symbols ,Sierpiński curve ,Complex plane ,Complex quadratic polynomial ,Mathematics - Abstract
In this paper we consider the family of rational maps of the complex plane given by z 2 + λ z 2 where λ is a complex parameter. We regard this family as a singular perturbation of the simple function z 2 . We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the correspondingmaps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However,we also showthatparameterscorrespondingto differentopensets havedynamics that are not conjugate.
- Published
- 2005
46. Commutators and diffeomorphisms of surfaces
- Author
-
Jean-Marc Gambaudo and Étienne Ghys
- Subjects
Surface (mathematics) ,Pure mathematics ,Dimension (vector space) ,Simple (abstract algebra) ,Homogeneous ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics ,Vector space - Abstract
For any compact oriented surfacewe consider the group of diffeomorphisms ofwhich preserve a given area form. In this paper we show that the vector space of homogeneous quasi-morphisms on this group has infinite dimension. This result is proved by constructing explicitly and for each surface an infinite family of independent homogeneous quasi-morphisms. These constructions use simple arguments related to linking properties of the orbits of the diffeomorphisms.
- Published
- 2004
47. Averaging principle for fully coupled dynamical systems and large deviations
- Author
-
Yuri Kifer
- Subjects
Slow motion ,Work (thermodynamics) ,Dynamical systems theory ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Calculus ,Large deviations theory ,Axiom A ,Variable (mathematics) ,Mathematics ,Hamiltonian system - Abstract
In the study of systems which combine slow and fast motions, the averaging principle suggests that a good approximation of the slow motion on long time intervals can be obtained by averaging its parameters over the fast variables. When the slow and fast motions depend on each other (fully coupled), as is usually the case, for instance, in perturbations of Hamiltonian systems, the averaging prescription cannot always be applied, and when it does work, this is usually only in some averaged with respect to initial conditions sense. In this paper we first give necessary and sufficient conditions for the averaging principle to hold (in the above sense) and then, relying on some large deviations arguments, verify them in the case when the fast motions are hyperbolic (Axiom A) flows for each freezed slow variable. It turns out that in this situation the Lebesgue measure of initial conditions with bad averaging approximation tends to zero exponentially fast as the parameter tends to zero.
- Published
- 2004
48. Lyapunov exponents and rates of mixing for one-dimensional maps
- Author
-
Stefano Luzzatto, José F. Alves, and Vilton Pinheiro
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Zero (complex analysis) ,Hölder condition ,Lyapunov exponent ,Absolute continuity ,symbols.namesake ,Exponential growth ,Mixing (mathematics) ,symbols ,Almost everywhere ,Invariant measure ,Mathematics - Abstract
We show that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive, some power of f is mixing and, in particular, the correlation of Holder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, by the average rate at which typical points start to exhibit exponential growth of the derivative.
- Published
- 2004
49. Ergodic sums of non-integrable functions under one-dimensional dynamical systems with indifferent fixed points
- Author
-
Tomoki Inoue
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Dynamical systems theory ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Absolute continuity ,Fixed point ,Ergodic theory ,Invariant measure ,Invariant (mathematics) ,Analytic function ,Mathematics - Abstract
We consider one-dimensional dynamical systems with indifferent fixed points (fixed points with derivative one). Many such maps have absolutely continuous ergodic infinite invariant measures. We study the limit of the ratio of the ergodic sum of f A to that of f B , where the integrals of f A and f B are infinite with respect to the absolutely continuous ergodic infinite invariant measure. If f A and f B are analytic functions on [0, 1], the result in this paper makes it clear whether the ratio of the ergodic sum of f A to that of f B converges in the Lebesgue measure or not.
- Published
- 2004
50. The explosion of singular-hyperbolic attractors
- Author
-
C. A. Morales
- Subjects
Rössler attractor ,Pure mathematics ,Mathematics::Dynamical Systems ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Hyperbolic manifold ,Lorenz system ,Stable manifold ,Nonlinear Sciences::Chaotic Dynamics ,Singularity ,Attractor ,Vector field ,Gravitational singularity ,Mathematics - Abstract
A singular-hyperbolic attractor for vector fields is a partially hyperbolic attractor with singularities (that are hyperbolic) and volume expanding central direction. The geometric Lorenz attractor is the most representative example of a singular-hyperbolic attractor. In this paper, we prove that if $\Lambda$ is a singular-hyperbolic attractor of a three-dimensional vector field X , then there is a neighborhood U of $\Lambda$ in M such that every attractor in U of a C r vector field C r close to X is singular, i.e. it contains a singularity. With this result we prove the following corollaries. There are neighborhoods U of $\Lambda$ (in M ) and $\mathcal U$ of X (in the space of C r vector fields) such that if n denotes the number of singularities of X in $\Lambda$ , then $\#\{A\subset U:A$ is an attractor of $Y\in\mathcal U\}\leq n$ . Every three-dimensional vector field C r close to one exhibiting a singular-hyperbolic attractor has a singularity non-isolated in the non-wandering set. A singularity of a three-dimensional C r vector field Y is stably non-isolated in the non-wandering set if it is the unique singularity of a singular-hyperbolic attractor of Y . These results generalize well-known properties of the geometric Lorenz attractor.
- Published
- 2004
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.