ATTRACTORS AND THEIR INVISIBLE PARTS FOR SKEW PRODUCTS WITH HIGH DIMENSIONAL FIBER

2012 ◽  
Vol 22 (08) ◽  
pp. 1250182 ◽  
Author(s):  
F. H. GHANE ◽  
M. NAZARI ◽  
M. SALEH ◽  
Z. SHABANI
Keyword(s):  
Compact Manifold ◽  
High Dimensional ◽  
Skew Product ◽  
Small Perturbations ◽  
Skew Products ◽  
Lipschitz Constants ◽  
Structurally Stable

In this article, we study statistical attractors of skew products which have an m-dimensional compact manifold M as a fiber and their ε-invisible subsets. For any n ≥ 100 m2, m = dim (M), we construct a set [Formula: see text] in the space of skew products over the horseshoe with the fiber M having the following properties. Each C2-skew product from [Formula: see text] possesses a statistical attractor with an ε-invisible part, for an extraordinary value of ε (ε = (m + 1)-n), whose size of invisibility is comparable to that of the whole attractor, and the Lipschitz constants of the map and its inverse are no longer than L. The set [Formula: see text] is a ball of radius O(n-3) in the space of skew products over the horseshoe with the C1-metric. In particular, small perturbations of these skew products in the space of all diffeomorphisms still have attractors with the same properties. Moreover, for skew products which have an m-sphere as a fiber, it consists of structurally stable skew products. Our construction develops the example of [Ilyashenko & Negut, 2010] to skew products which have an m-dimensional compact manifold as a fiber, m ≥ 2.

scholarly journals Nesting Monte Carlo for high-dimensional non-linear PDEs

2018 ◽  
Vol 24 (4) ◽  
pp. 225-247 ◽  
Author(s):  
Xavier Warin
Keyword(s):  
Monte Carlo ◽  
Deep Learning ◽  
High Dimension ◽  
Analytical Solutions ◽  
New Method ◽  
Computational Time ◽  
High Dimensional ◽  
Learning Methods ◽  
Lipschitz Constants ◽  
Linear Pdes

Abstract A new method based on nesting Monte Carlo is developed to solve high-dimensional semi-linear PDEs. Depending on the type of non-linearity, different schemes are proposed and theoretically studied: variance error are given and it is shown that the bias of the schemes can be controlled. The limitation of the method is that the maturity or the Lipschitz constants of the non-linearity should not be too high in order to avoid an explosion of the computational time. Many numerical results are given in high dimension for cases where analytical solutions are available or where some solutions can be computed by deep-learning methods.

scholarly journals Smale endomorphisms over graph-directed Markov systems

2020 ◽  
pp. 1-34
Author(s):  
EUGEN MIHAILESCU ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Equilibrium Measure ◽  
Parabolic Systems ◽  
Skew Product ◽  
Pointwise Dimension ◽  
Markov Systems ◽  
Large Classes ◽  
Equilibrium Measures ◽  
Skew Products ◽  
Natural Extensions ◽  
Induced Maps

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$ -maps $T_{\unicode[STIX]{x1D6FD}}$ , for arbitrary $\unicode[STIX]{x1D6FD}>1$ .

scholarly journals Lipeomorphisms close to an Anosov diffeomorphism

1974 ◽  
Vol 53 ◽  
pp. 71-82 ◽  
Author(s):  
Kentaro Takaki
Keyword(s):  
Compact Manifold ◽  
Anosov Diffeomorphism ◽  
Structurally Stable

It is well-known that an Anosov diffeomorphism f on a compact manifold is structurally stable in the space of all C1-diffeomorphisms, with the C1-topology (Anosov [1]). In this paper we show that f is also structurally stable in the space of all lipeomorphisms, with a lipschitz topology. The proof is similar to that of the C1-case by J. Moser [4].

scholarly journals Bistable vector fields are axiom A

1995 ◽  
Vol 51 (1) ◽  
pp. 83-86
Author(s):  
Mike Hurley
Keyword(s):  
Vector Field ◽  
Compact Manifold ◽  
Vector Fields ◽  
Axiom A ◽  
Smooth Compact Manifold ◽  
Structurally Stable

Recently L. Wen showed that if a C1 vector field (on a smooth compact manifold without boundary) is both structurally stable and topologically stable then it will satisfy Axiom A. The purpose of this note is to indicate how results from an earlier paper can be used to simplify somewhat Wen's argument.

Efficient Numerical Shadowing Global Error Estimation for High Dimensional Dissipative Systems

2004 ◽  
Vol 4 (2) ◽  
Author(s):  
Jon Collis ◽  
Erik S. Van Vleck
Keyword(s):  
Error Estimation ◽  
Initial Conditions ◽  
Efficient Estimation ◽  
Global Error ◽  
Dissipative Systems ◽  
High Dimensional ◽  
Initial Value ◽  
Small Perturbations ◽  
One Step ◽  
Global Error Estimation

AbstractShadowing is a means of characterizing global errors in the numerical solution of initial value differential equations by allowing for small perturbations in the initial conditions. The method presented in this paper provides a technique for efficient estimation of the shadowing global error for systems that have a large number of exponentially decaying modes. The method is formulated for one-step methods and is applied to the spatial discretization of some dissipative PDEs.

scholarly journals Robust Density of Periodic Orbits for Skew Products with High Dimensional Fiber

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Fatemeh Helen Ghane ◽  
Mahboubeh Nazari ◽  
Mohsen Saleh ◽  
Zahra Shabani
Keyword(s):  
Periodic Orbits ◽  
High Dimensional ◽  
Skew Products

The Core and the Stability of Group Choice in Spatial Voting Games

1988 ◽  
Vol 82 (1) ◽  
pp. 195-211 ◽  
Author(s):  
Norman Schofield ◽  
Bernard Grofman ◽  
Scott L. Feld
Keyword(s):  
Political Stability ◽  
Simple Majority ◽  
Small Perturbations ◽  
The Core ◽  
Spatial Voting ◽  
Voting Games ◽  
Voting Game ◽  
Stable Core ◽  
The Stability ◽  
Structurally Stable

The core of a voting game is the set of undominated outcomes, that is, those that once in place cannot be overturned. For spatial voting games, a core is structurally stable if it remains in existence even if there are small perturbations in the location of voter ideal points. While for simple majority rule a core will exist in games with more than one dimension only under extremely restrictive symmetry conditions, we show that, for certain supramajorities, a core must exist. We also provide conditions under which it is possible to construct a structurally stable core. If there are only a few dimensions, our results demonstrate the stability properties of such frequently used rules as two-thirds and three-fourths. We further explore the implications of our results for the nature of political stability by looking at outcomes in experimental spatial voting games and at Belgian cabinet formation in the late 1970s.

scholarly journals Small C1-smooth perturbations of skew products and the partial integrability property

2020 ◽  
Vol 5 (2) ◽  
pp. 317-328
Author(s):  
L.S. Efremova
Keyword(s):  
Sufficient Conditions ◽  
Skew Product ◽  
Interval Maps ◽  
Skew Products ◽  
Partial Integrability

AbstractIn this paper we investigate stability of the integrability property of skew products of interval maps under small C1-smooth perturbations satisfying some conditions. We obtain here (sufficient) conditions of the partial integrability for maps under considerations. These conditions are formulated in the terms of properties of the unperturbed skew product. We give also the example of the partially integrable map.

scholarly journals A counterexample to a positive entropy skew product generalization of the Pinsker conjecture

1985 ◽  
Vol 5 (3) ◽  
pp. 379-407
Author(s):  
Jonathan King
Keyword(s):  
Positive Entropy ◽  
Skew Product ◽  
Skew Products ◽  
Fibre Transformation

AbstractThe class of k-automorphisms is not contained in a certain class of skew products over a Bernoulli base. The non-identity fibre transformation in the skew is allowed to have positive or even infinite entropy. A difficulty presented by positive entropy is handled via an apparently new property of independent processes (lemma 7.24).

A metric minimal PI cascade with minimal ideals

2018 ◽  
Vol 40 (5) ◽  
pp. 1268-1281
Author(s):  
ELI GLASNER ◽  
YAIR GLASNER
Keyword(s):  
Compact Manifold ◽  
Minimal Flow ◽  
Connected Group ◽  
Skew Products ◽  
Enveloping Semigroup

We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than $2^{\mathfrak{c}}$ (where $\mathfrak{c}=2^{\aleph _{0}}$) minimal left ideals is proximal isometric (PI). Then we show the existence of various minimal PI-flows with many minimal left ideals, as follows. For the acting group $G=\text{SL}_{2}(\mathbb{R})^{\mathbb{N}}$, we construct a metric minimal PI $G$-flow with $\mathfrak{c}$ minimal left ideals. We then use this example and results established in Glasner and Weiss. [On the construction of minimal skew-products. Israel J. Math.34 (1979), 321–336] to construct a metric minimal PI cascade $(X,T)$ with $\mathfrak{c}$ minimal left ideals. We go on to construct an example of a minimal PI-flow $(Y,{\mathcal{G}})$ on a compact manifold $Y$ and a suitable path-wise connected group ${\mathcal{G}}$ of a homeomorphism of $Y$, such that the flow $(Y,{\mathcal{G}})$ is PI and has $2^{\mathfrak{c}}$ minimal left ideals. Finally, we use this latter example and a theorem of Dirbák to construct a cascade $(X,T)$ that is PI (of order three) and has $2^{\mathfrak{c}}$ minimal left ideals. Thus this final result shows that, even for cascades, the converse of the implication ‘less than $2^{\mathfrak{c}}$ minimal left ideals implies PI’, fails.

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