scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

scholarly journals Lyapunov spectrum for Hénon-like maps at the first bifurcation

2016 ◽  
Vol 38 (3) ◽  
pp. 1168-1200
Author(s):  
HIROKI TAKAHASI
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Level Sets ◽  
Multifractal Analysis ◽  
Lyapunov Spectrum ◽  
Probability Measures ◽  
Limit Set ◽  
Uniform Hyperbolicity ◽  
Partial Description ◽  
Set Of Points

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.

scholarly journals The Geometry of the Weak Lefschetz Property and Level Sets of Points

2008 ◽  
Vol 60 (2) ◽  
pp. 391-411 ◽  
Author(s):  
Juan C. Migliore
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Hilbert Function ◽  
Worst Case ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Set Of Points

AbstractIn a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

scholarly journals A relation between Lyapunov exponents, Hausdorff dimension and entropy

1981 ◽  
Vol 1 (4) ◽  
pp. 451-459 ◽  
Author(s):  
Anthony Manning
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Unstable Manifold ◽  
Positive Lyapunov Exponent ◽  
Axiom A ◽  
Generic Points

AbstractFor an Axiom A diffeomorphism of a surface with an ergodic invariant measure we prove that the entropy is the product of the positive Lyapunov exponent and the Hausdorff dimension of the set of generic points in an unstable manifold.

The topological entropy of non-dense orbits and generalized Schmidt games

2017 ◽  
Vol 39 (2) ◽  
pp. 500-530
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Topological Entropy ◽  
Invariant Measures ◽  
Metric Entropy ◽  
Toral Automorphism ◽  
Partially Hyperbolic ◽  
Dense Orbits ◽  
Bounded Below ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism

We generalize the notion of Schmidt games to the setting of the general Caratheódory construction. The winning sets for such generalized Schmidt games usually have large corresponding Caratheódory dimensions (e.g., Hausdorff dimension and topological entropy). As an application, we show that for every $C^{1+\unicode[STIX]{x1D703}}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ satisfying certain technical conditions, the topological entropy of the set of points with non-dense forward orbits is bounded below by the unstable metric entropy (in the sense of Ledrappier–Young) of certain invariant measures. This also gives a unified proof of the fact that the topological entropy of such a set is equal to the topological entropy of $f$, when $f$ is a toral automorphism or the time-one map of a certain non-quasiunipotent homogeneous flow.

scholarly journals Invariant measures for interval maps without Lyapunov exponents

2021 ◽  
pp. 1-29
Author(s):  
JORGE OLIVARES-VINALES
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Critical Points ◽  
Invariant Measures ◽  
Full Measure ◽  
Interval Maps ◽  
Interval Map

Abstract We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.

scholarly journals Multifractal analysis in non-uniformly hyperbolic interval maps

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 110-133
Author(s):  
Guanzhong Ma ◽  
Wenqiang Shen ◽  
Xiao Yao
Keyword(s):  
Hausdorff Dimension ◽  
Level Set ◽  
Multifractal Analysis ◽  
Ergodic Measure ◽  
Moran Set ◽  
Interval Maps ◽  
The Given

Abstract In this paper, we establish a framework for the construction of Moran set driven by dynamics. Under this framework, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic measure in a class of non-uniformly hyperbolic interval maps with finitely many branches.

Dynamical upper bounds for Hausdorff dimension of invariant sets

1997 ◽  
Vol 17 (3) ◽  
pp. 739-756 ◽  
Author(s):  
YINGJIE ZHANG
Keyword(s):  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Upper Bounds ◽  
Invariant Sets ◽  
Topological Pressure ◽  
Expanding Maps ◽  
Unstable Manifolds ◽  
Hyperbolic Sets

We study the Hausdorff dimension of invariant sets for expanding maps and that of hyperbolic sets on unstable manifolds. Upper bounds for the Hausdorff dimension are given in terms of topological pressure, or topological entropy and Lyapunov exponents.

scholarly journals On shrinking targets for piecewise expanding interval maps

2015 ◽  
Vol 37 (2) ◽  
pp. 646-663 ◽  
Author(s):  
TOMAS PERSSON ◽  
MICHAŁ RAMS
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Gibbs Measure ◽  
Intersection Property ◽  
Interval Maps ◽  
Large Intersection ◽  
Set Of Points

For a map $T:[0,1]\rightarrow [0,1]$ with an invariant measure $\unicode[STIX]{x1D707}$, we study, for a $\unicode[STIX]{x1D707}$-typical $x$, the set of points $y$ such that the inequality $|T^{n}x-y|<r_{n}$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D719}}$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

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