Volume lemmas and large deviations for partially hyperbolic endomorphisms

2019 ◽  
Vol 41 (1) ◽  
pp. 213-240
Author(s):  
ANDERSON CRUZ ◽  
GIOVANE FERREIRA ◽  
PAULO VARANDAS
Keyword(s):  
Lebesgue Measure ◽  
Large Deviation ◽  
Positive Lebesgue Measure ◽  
Stable Bundle ◽  
Srb Measure ◽  
Cone Field ◽  
Mild Assumption ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic ◽  
Set Of Points

We consider partially hyperbolic attractors for non-singular endomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. We prove volume lemmas for both Lebesgue measure on the topological basin of the attractor and the SRB measure supported on the attractor. As a consequence, under a mild assumption we prove exponential large-deviation bounds for the convergence of Birkhoff averages associated to continuous observables with respect to the SRB measure.

scholarly journals SRB measures for partially hyperbolic attractors of local diffeomorphisms

2018 ◽  
Vol 40 (6) ◽  
pp. 1545-1593
Author(s):  
ANDERSON CRUZ ◽  
PAULO VARANDAS
Keyword(s):  
Thermodynamic Formalism ◽  
Positive Lebesgue Measure ◽  
Stable Bundle ◽  
Local Diffeomorphism ◽  
Local Diffeomorphisms ◽  
Axiom A ◽  
Srb Measures ◽  
Cone Field ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local diffeomorphism.

RIDDLED BASINS AND UNSTABLE DIMENSION VARIABILITY IN CHAOTIC SYSTEMS WITH AND WITHOUT SYMMETRY

2001 ◽  
Vol 11 (10) ◽  
pp. 2689-2698 ◽  
Author(s):  
RICARDO L. VIANA ◽  
CELSO GREBOGI
Keyword(s):  
Lebesgue Measure ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Symmetric Case ◽  
Positive Lebesgue Measure ◽  
Two Dimensional ◽  
Symmetry Breaking Term ◽  
Dissipative Dynamical Systems ◽  
Set Of Points ◽  
Breaking Term

Riddling occurs in dissipative dynamical systems with more than one attractor, when the basin of one attractor is punctured with holes belonging to the basins of the other attractors. The basin of a chaotic attractor is riddled if (i) it has a positive Lebesgue measure; (ii) in the vicinity of every point belonging to the basin of the attractor, there is a positive Lebesgue measure set of points that asymptote to another attractor. We investigate the presence of riddled basins in a two-dimensional noninvertible map with a symmetry-breaking term. In the symmetric case the onset of riddling is characterized by an unstable–unstable pair bifurcation, which also leads to unstable dimension variability in the invariant chaotic set. The nonsymmetric case exhibits a chaotic attractor, but a riddled basin occurs only at the bifurcation point, since after that the attractor becomes a chaotic saddle. We analyze the presence of unstable dimension variability in the symmetric case by computing the finite-time transverse Lyapunov exponents. We point out some consequences of those facts to the synchronization properties of coupled chaotic systems.

scholarly journals Fundamental domain of invariant sets and applications

2012 ◽  
Vol 34 (1) ◽  
pp. 341-352 ◽  
Author(s):  
PENGFEI ZHANG
Keyword(s):  
Fundamental Domain ◽  
Sufficient Conditions ◽  
Invariant Sets ◽  
Positive Lebesgue Measure ◽  
Positive Volume ◽  
Transitive Set ◽  
Partially Hyperbolic ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism ◽  
Full Volume

AbstractLet $X$ be a compact metric space, $f:X\to X$ a homeomorphism and $\phi \in C(X,\mathbb {R})$. We construct a fundamental domain for the set of points with finite peaks with respect to the induced cocycle $\{\phi _n\}$. As applications, we give sufficient conditions for the transitive set of a non-conservative partially hyperbolic diffeomorphism to have positive Lebesgue measure, i.e., for an accessible partially hyperbolic diffeomorphism, if the set of points with finite peaks for the Jacobian cocycle is not of full volume, then the set of transitive points is of positive volume.

scholarly journals Physical measures for certain partially hyperbolic attractors on 3-manifolds

2017 ◽  
Vol 39 (1) ◽  
pp. 74-104 ◽  
Author(s):  
RICARDO T. BORTOLOTTI
Keyword(s):  
Finite Number ◽  
Lebesgue Measure ◽  
Gibbs States ◽  
Stable Foliation ◽  
Ergodic Properties ◽  
Central Direction ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic ◽  
Physical Measures ◽  
Full Lebesgue Measure

In this work, we analyze ergodic properties of certain partially hyperbolic attractors whose central direction has a neutral behavior; the main feature is a condition of transversality between the projections of unstable leaves, projecting through the stable foliation. We prove that partial hyperbolic attractors satisfying this condition of transversality, neutrality in the central direction and regularity of the stable foliation admit a finite number of physical measures, coinciding with the ergodic u-Gibbs States, whose union of the basins has full Lebesgue measure. Moreover, we describe the construction of robustly non-hyperbolic attractors satisfying these properties.

Strange Nonchaotic Trajectories on Torus

1997 ◽  
Vol 07 (02) ◽  
pp. 423-429 ◽  
Author(s):  
T. Kapitaniak ◽  
L. O. Chua
Keyword(s):  
Lebesgue Measure ◽  
Parameter Space ◽  
Positive Lebesgue Measure ◽  
Strange Nonchaotic Attractors

In this letter we have shown that aperiodic nonchaotic trajectories characteristic of strange nonchaotic attractors can occur on a two-frequency torus. We found that these trajectories are robust as they exist on a positive Lebesgue measure set in the parameter space.

scholarly journals Almost sure rates of mixing for partially hyperbolic attractors

2022 ◽  
Vol 311 ◽  
pp. 98-157
Author(s):  
José F. Alves ◽  
Wael Bahsoun ◽  
Marks Ruziboev
Keyword(s):  
Hyperbolic Attractors ◽  
Partially Hyperbolic

Simultaneous dense and non-dense orbits for certain partially hyperbolic diffeomorphisms

2018 ◽  
Vol 40 (4) ◽  
pp. 1083-1107
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Space ◽  
Anosov Diffeomorphism ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Dense Orbits ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism ◽  
Stable Subspace

Let$g:M\rightarrow M$be a$C^{1+\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on$M$. We show that, if$f:M\rightarrow M$is a$C^{1+\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of$f$and$g$span the whole tangent space at some point on$M$, the set of points that equidistribute under$g$but have non-dense orbits under$f$has full Hausdorff dimension. The same result is also obtained when$M$is the torus and$f$is a toral endomorphism whose center-stable subspace does not contain the stable subspace of$g$at some point.

On Convex Fundamental Regions for a Lattice

1961 ◽  
Vol 13 ◽  
pp. 177-178 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Lebesgue Measure ◽  
Convex Set ◽  
Fundamental Region ◽  
Linearly Independent ◽  
Whole Space ◽  
Closed Convex Set ◽  
Set Of Points

Let ∧ be a lattice in Euclidean n-space, that is, ∧ is a set of points ε1a1+ … + εnanwherea1… ,anare linearly independent vectors and the ε run over all integers. Letμdenote the Lebesgue measure. A closed convex setFis called afundamental regionfor ∧ if the setsF+x (x∈∧) cover the whole space without overlapping; that is, ifF0is the interior ofF,and 0 ≠x∈ ∧, thenF0∩(F0+x) =ϕ.

scholarly journals Torsion Discriminance for Stability of Linear Time-Invariant Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 386
Author(s):  
Yuxin Wang ◽  
Huafei Sun ◽  
Yueqi Cao ◽  
Shiqiang Zhang
Keyword(s):  
Lebesgue Measure ◽  
Linear Time ◽  
Zero Solution ◽  
Positive Lebesgue Measure ◽  
Asymptotically Stable ◽  
Linear Time Invariant ◽  
Measurable Set ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

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