All Questions
Tagged with hyperbolic-dynamics ds.dynamical-systems
59 questions
2
votes
1
answer
256
views
Chaotic dynamics of maps on unit square that are NOT Triangular
We will denote the compact interval $[0,1]$ by $I$ and the unit square $[0,1]\times[0,1]$ by $I^2$. Triangular map on $I^2$ is a continuous map $F:I^2\to I^2$ of the form $F(x,y)=(f(x),g(x,y))$ where $...
-2
votes
1
answer
210
views
Reference request on dynamics and hyperbolic dynamics (hyperbolicity in absence of periodic orbits)
I would appreciate if you introduce me a reference (paper or book) who address the concept of hyperbolic dynamics but with emphasis on absence of periodic orbits. a possible ...
- 6,581
0
votes
0
answers
36
views
Some equivalent conditions for hyperbolicity of flow
Let $M$ be a manifold and $\phi_t$ be a smooth flow associated to a smooth vector field on $M$.
Are the following 3 conditions equivalent?
1)For every fixed $t$ the diffeomorphism $\...
- 356
4
votes
1
answer
180
views
Fixed points of maps defined on Teichmüller space
Let $\mathcal{T}_A$ be a Teichmüller space of the sphere $S^2$ with a finite set $A$ of marked points, and suppose that $f \colon \mathcal{T}_A \to \mathcal{T}_A$ is a holomorphic map that has a ...
- 12.3k
8
votes
0
answers
156
views
Square root of an Anosov diffeomorphism
Let $T\colon \mathbb T^d\to\mathbb T^d$ be an Anosov diffeomorphism (that is, the tangent bundle splits into invariant stable and unstable bundles; the restriction of $DT$ to the unstable bundle is ...
- 23.2k
1
vote
0
answers
67
views
Equidistribution on subvarieties of $\mathrm{SL}_n(\mathbb{Z})$
$\DeclareMathOperator\SL{SL}$Let $\Gamma_{\infty}$ be a maximal parabolic subgroup of $\SL_n(\mathbb{Z})$. I believe there are various equidistribution theorems, which say that Iwasawa $x$ coordinates ...
- 63.9k
2
votes
0
answers
319
views
A (possible) generic spectral property in one dimensional dynamics
Context and Definitions
Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if:
$T$ has a finite number of hyperbolic periodic attractors; and
defining $...
- 333
3
votes
1
answer
81
views
Can the Reeb foliation of $S^3$ be realized as stable manifold foliation of a smooth hyperbolic discrete dynamic on $S^3$?
Can the Reeb foliation of $S^3$ be realized as foliation associated to stable(or unstable) manifolds of a hyperbolic discrete dynamic on $S^3$?If yes what is a precise formulation for that ...
- 6,581
1
vote
1
answer
163
views
Existence of center-stable manifold when the Jacobian is singular?
The following is a result from Shub's monograph "Global Stability of Dynamical Systems".
I dabble in the proof, and it appears to me that the existence of $W^{\rm cu}_{\rm loc}$ does not ...
- 798
6
votes
4
answers
763
views
A follow up question related to entropy
For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...
- 1,446
3
votes
0
answers
71
views
Trapped vs. nonwandering points
For a continuous flow on a topological space the concept of a nonwandering point (in forward/backward time) is well-known. Another useful concept (for example from the point of view of scattering ...
- 1,942
2
votes
1
answer
187
views
How to find the hyperbolic dimension of map $f(z) = z^2$ of $\overline{\mathbb{C}}$ onto itself?
I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. In his article, he defined hyperbolic sets and hyperbolic ...
- 45
2
votes
0
answers
83
views
Question about stable manifold theorem and Frobenius integrability theorem
I have a question about Anosov diffeomorphism (Wikipedia: Anosov diffeomorphisms)
For hyperbolic fixed point $p$, $W^{s}(p)$ is a smooth manifold and its tangent space has the same dimension as the ...
- 63.9k
2
votes
1
answer
122
views
Examples of hyperbolic set and J-stable sets
I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. The following two definitions are given without any examples in ...
- 6,548
0
votes
0
answers
103
views
Polynomial / quadratic autonomous system of ODEs – proving monotonicity / convexity
Problem:
Consider the autonomous ODE system
\begin{align*}
\dot{x} &= (1-x) (z-xy)\\
\dot{y} &= \tfrac 1 2 y^2 - (a+xy)(1-y) \\
\dot{z} &= \tfrac 1 2 z^2 - \tfrac 1 2 y^2 + (a+xy)z
\end{...
- 291
6
votes
0
answers
348
views
Examples of expansive homeomorphisms with the specification property that are neither symbolic nor factors of mixing SFT nor product of thereof
I am looking for nontrivial examples of expansive homeomorphisms with the specification property on compact metric spaces. Here, by a ``trivial'' example I understand a subshift with the specification ...
- 1,658
26
votes
7
answers
2k
views
If you were to axiomatize the notion of entropy
What are the axioms that a good notion of entropy must satisfy? Please note that I am not asking for the definitions of various types of entropy such as topological entropy or measure-theoretic ...
CommunityBot
- 1
2
votes
0
answers
76
views
Periodic orbits of generalized cat map near the origin
Let $M\in SL(2,\mathbb{Z})$ have eigenvalues $\lambda, \lambda^{-1}$ with $\lambda>1$., and suppose $M$ is diagonalized as $Q\Lambda Q^{-1}$ with $\det Q=1$. (Note that his doesn't determine $Q$, ...
- 854
1
vote
0
answers
91
views
Random matrix heuristics for Koopman operators
Consider a nice hyperbolic dynamical system $(X, T)$, for instance a $\mathcal{C}^\infty$ Anosov map. The action of the Koopman operator
$$\mathcal{K} : \ f \mapsto f \circ T$$
has a nice spectrum ...
- 581
0
votes
0
answers
92
views
Homoclinically related hyperbolic periodic points gives the same pesin homoclinic class up to null sets
In MINIMALITY AND STABLE BERNOULLINESS IN DIMENSION 3 by Nunez and Hertz, the first paragraph in the proof of Corollary 2.4 says the above statement follows by using a "$\lambda$-lemma". ...
- 11
2
votes
0
answers
95
views
Persistence of homoclinic points in the non-compact case
It is well known that a transverse homoclinic point of a hyperbolic periodic point of a $C^1$-diffeomorphism of a compact manifold $M$ persists under small $C^1$ perturbations. This follows easily ...
- 111
1
vote
0
answers
175
views
Example of topologically transitive dynamical system with invariant non-ergodic Borel measure
Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which
$f : \Lambda \to \...
- 151
2
votes
0
answers
48
views
Whether or not two distinct points in Teichmuller space induce absolutely continuous volume forms on the unit tangent bundle of a surface?
Let $S$ be a closed orientable surface of genus greater than two. Let $g$ and $g'$ be metrics two of constant curvature. I guess we an think of these as two points in the Teichmüller space $\mathcal{T}...
- 401
2
votes
0
answers
83
views
Center-stable manifold theorem on manifold with boundary
I would like to see if there is a Center-stable manifold theorem on the phase space that is a manifold with boundary.
Suppose $f:M\rightarrow M$ is a diffeomorphism, according to Theorem III.7 in "...
- 21
1
vote
1
answer
164
views
Example of zero Lyapunov exponentes
Assume that $(T, A)$ is a linear cocycle such that $T:X\rightarrow X$ is a homemorphism on compact metric space $X$ and $A:X\rightarrow SL(2, \mathbb{R})$ is a continuous function.
We say that an ...
- 1,043
0
votes
0
answers
61
views
Unique poine in holonomies
Let $\Lambda$ be Axiom A for $C^{1+\gamma}$ $f$. I am reading this paper. I have a problem to undestand holonomies. The holonomy mapping
$$ h: W_{loc}^{s} (x) \cap\Lambda \rightarrow W_{loc}^{s} (y) \...
- 1,043
2
votes
0
answers
124
views
On invariant cones of the Katok map
I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
- 151
4
votes
1
answer
212
views
When entropy SRB measure is zero
It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures.
Let $f:M \rightarrow ...
- 8,903
4
votes
2
answers
729
views
Centre manifold theory for a curve of equilibrium points
I am looking for advice concerning a specific situation related to centre manifold theory (compare Perko 2001).
The part which is known
Let's consider a differential equation in higher-dimensional ...
CommunityBot
- 1
4
votes
1
answer
246
views
Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$
Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
- 199
1
vote
0
answers
53
views
Limit contration rates and expansion rate solenoid map
Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...
3
votes
1
answer
360
views
Question on a proof of density of periodic orbits
In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following:
Theorem: Let $\Gamma$ be a ...
- 23.2k
4
votes
1
answer
132
views
The continuity of the the stable and unstable in definition of hyperbolic sets for flows
I would like to know whether the continuity of the stable and unstable subbundles $E^{s}$ and $E^{u}$ follows from the growth conditions as in the discrete case, or must be hypothesized, in the ...
- 3,928
4
votes
1
answer
536
views
Teichmueller disk and the $\mathrm{SL}_2\mathbb{R}$ action
Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}_g$ be the space of holomorphic 1-forms over genus $...
- 41
3
votes
0
answers
134
views
Why is a hyperbolic basic set of dimension 2 either an attractor or a repeller?
I'm currently trying to understand the Birman-Williams Template Theorem, proved in the paper "Knotted periodic orbits II: Fibered knots, Low Dimensional Topology". Unfortunately, there doesn't seem to ...
- 318
7
votes
1
answer
281
views
Quantitative approximation of invariant measures by periodic ones
It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures. Is there some knowledge ...
- 2,479
2
votes
0
answers
94
views
Algebraic approach to prove the mixing property of Lorenz flow on hyperbolic surface
We knew that the noncompact subgroups of SL(2,$\mathbb{R}$) are mixing by Howe-Moore ergodicity theorem. I am curious about Lorenz flow, if we have a algebraic approach to prove the mixing property of ...
- 53
5
votes
1
answer
988
views
Ergodicity and mixing of geodesic and horocyclic flows
I am reading a theorem about the ergodicity and mixing of geodesic and horocyclic flows on unit tangent bundle of a compact hyperbolic surface. I find that there are two ways (be listed below) to ...
- 18.7k
1
vote
1
answer
128
views
Continuity of Lyapunov spaces
The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989:
Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let $f\in Diff^{1+\...
- 485
3
votes
0
answers
133
views
If $f$ is dynamically coherent, is there a unique invariant foliation tangent to $E^{c}$?
Let $f$ be a diffeomorphism of a closed manifold $M$ such that $f$ is partially hyperbolic if the tanget bundle of $M$, $TM$ splits into three invariant sub-bunbles
$$
TM = E^{s} \oplus E^{c} \oplus ...
CommunityBot
- 1
1
vote
0
answers
51
views
About stable manifold of a point [closed]
Let $(X, d)$ be a compact metric space and $f:X\rightarrow X$ be a homeomorphism and
$$W^{s}(x)=\{y| d(f^{n}(x), f^{n}(y))\rightarrow as \ n\rightarrow \infty\}.$$
Question: What condition on $(X, ...
- 121
13
votes
1
answer
775
views
Applications of the Central Limit Theorem in dynamical systems
There are very many papers in the area of (possibly non-uniformly) hyperbolic dynamical systems whose aim is to prove the Central Limit Theorem. In a dynamical context, this means that one:
has a ...
- 326
3
votes
3
answers
1k
views
reference on complex dynamics
Please someone suggest me some reference on the topic "Complex Dynamics". I want a brief geometric treatment from the root level. I have graduate level background on complex analysis, riemannian ...
- 3,709
2
votes
0
answers
283
views
Uniqueness of analytic center manifold
In a book, i have read a remark which says that the center manifold of an equilibrium point of a differential equation is not unique in general but is unique in the class of analytic manifold. The ...
- 2,481
0
votes
0
answers
107
views
Asymptotic pseudo orbit of an action
Let $G$ be finitely generated group (i.e $G= <S>$ $S=\{ s_1, ...,
s_n\}$) and $\varphi:G\times M\longrightarrow M$ is an action then
$f:G\longrightarrow M$ is called $\delta$- pseudo orbit if $...
5
votes
0
answers
244
views
introduction books for Dynamic systems of discrete Schrodinger operator for beginner
In this semester, I study in a class of dynamic system. recently the French professor turn to the dynamic system of discrete operator. I find it is difficult to find a book in English. (I have found ...
CommunityBot
- 1
1
vote
2
answers
433
views
Extension of integrable distribution over a subset
Let $M$ be a smooth manifold and $G_k(M)$ be the $k$-dimensional Grassmian bundle of $M$. Let $K\subset M$ be a compact subset and $E:K\to G_k(M)$ be a continuous distribution on $K$.
We say $E$ is ...
- 31
8
votes
1
answer
652
views
Lebesgue entropy zero and positive topological entropy
I am looking for examples of volume preserving $C^{\infty}$ diffeomorphisms $f$ of a surface, which have positive topological entropy ($h(f) > 0$), but that the Lebesgue measure entropy (metric ...
- 2,244
2
votes
1
answer
752
views
Angle between two subspaces
Let $f:M\to M$ be a diffeomorphism on a compact riemannian manifold $M$.In the definition of a hyperbolic set we know that for all $x\in M$ there is a splitting of tangent space
$T_xM=E^s(x)\oplus E^...
- 2,481
2
votes
1
answer
283
views
the union of local stable manifolds along local unstable manifolds
Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s_\delta(x), W^...
- 987