Invariant probability measure of circle maps with breaks and Hausdorff dimension

2020 ◽  
Vol 2020 (2) ◽  
pp. 4-15
Author(s):  
A. Aliyev
Keyword(s):  
Hausdorff Dimension ◽  
Probability Measure ◽  
Invariant Probability Measure ◽  
Circle Maps

Bounded Geometry in the Supports of Ergodic Invariant Probability Measures

1998 ◽  
Vol 08 (10) ◽  
pp. 1957-1973 ◽  
Author(s):  
Jun Hu
Keyword(s):  
Hausdorff Dimension ◽  
Periodic Orbit ◽  
Probability Measure ◽  
Lebesgue Measure ◽  
Turning Points ◽  
Invariant Probability Measure ◽  
Cantor Set ◽  
Probability Measures ◽  
Bounded Geometry

In this paper we put some techniques and methods of [Misurewicz, 1979; Hu & Sullivan, 1997; Hu & Tresser, 1998; Blokh & Lyubich, 1990; Martens et al., 1992] together to show that if a map f, from an interval I into itself with finitely many turning points, satisfies a new smooth regularity in [Hu & Sullivan, 1997] and is on the boundary of chaos, and if μ is an ergodic f-invariant probability measure on I which is not concentrated on a periodic orbit of f, then the support K of μ is a Cantor set of bounded geometry, and hence has Lebesgue measure 0 and Hausdorff dimension strictly between 0 and 1. We also include some natural examples which satisfy this new smooth regularity rather than the traditional ones.

scholarly journals Ergodic properties of bimodal circle maps

2017 ◽  
Vol 39 (06) ◽  
pp. 1462-1500
Author(s):  
SYLVAIN CROVISIER ◽  
PABLO GUARINO ◽  
LIVIANA PALMISANO
Keyword(s):  
Probability Measure ◽  
Periodic Orbits ◽  
Invariant Probability Measure ◽  
Physical Measure ◽  
Absolutely Continuous ◽  
Irrational Numbers ◽  
Circle Maps ◽  
Rotation Interval ◽  
Ergodic Properties ◽  
Absolutely Continuous Invariant

We give conditions that characterize the existence of an absolutely continuous invariant probability measure for a degree one $C^{2}$ endomorphism of the circle which is bimodal, such that all its periodic orbits are repelling, and such that both boundaries of its rotation interval are irrational numbers. Those conditions are satisfied when the boundary points of the rotation interval belong to a Diophantine class. In particular, they hold for Lebesgue almost every rotation interval. By standard results, the measure obtained is a global physical measure and it is hyperbolic.

scholarly journals On uniformly distributed orbits of certain horocycle flows

1982 ◽  
Vol 2 (2) ◽  
pp. 139-158 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Periodic Point ◽  
Open Subset ◽  
Invariant Probability Measure ◽  
Zero Measure ◽  
Lebesque Measure ◽  
Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

scholarly journals Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

2019 ◽  
Vol 23 ◽  
pp. 797-802
Author(s):  
Raphaël Cerf ◽  
Joseba Dalmau
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Branching Process ◽  
Invariant Probability Measure ◽  
Classical Formula ◽  
Multitype Branching Process ◽  
Primitive Matrix ◽  
Watson Process

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

Invariant measures for interval maps with different one-sided critical orders

2013 ◽  
Vol 35 (3) ◽  
pp. 835-853 ◽  
Author(s):  
HONGFEI CUI ◽  
YIMING DING
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Mild Condition ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Jump Discontinuities ◽  
Point Set ◽  
Invent Math

AbstractFor an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

scholarly journals Dimension, entropy and Lyapunov exponents

1982 ◽  
Vol 2 (1) ◽  
pp. 109-124 ◽  
Author(s):  
Lai-Sang Young
Keyword(s):  
Hausdorff Dimension ◽  
Probability Measure ◽  
Lyapunov Exponents ◽  
Borel Probability Measure ◽  
Fractional Dimension ◽  
Present Context ◽  
Rényi Dimension ◽  
Borel Probability

AbstractWe consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.

Continuity of Lyapunov exponents for random two-dimensional matrices

2016 ◽  
Vol 37 (5) ◽  
pp. 1413-1442 ◽  
Author(s):  
CARLOS BOCKER-NETO ◽  
MARCELO VIANA
Keyword(s):  
Probability Measure ◽  
Lyapunov Exponents ◽  
Invariant Probability Measure ◽  
Two Dimensional ◽  
Bernoulli Shifts

The Lyapunov exponents of locally constant$\text{GL}(2,\mathbb{C})$-cocycles over Bernoulli shifts vary continuously with the cocycle and the invariant probability measure.

Peculiarities of the relaxation to an invariant probability measure of nonhyperbolic chaotic attractors in the presence of noise

2002 ◽  
Vol 65 (3) ◽  
Author(s):  
Vadim S. Anishchenko ◽  
Tatjana E. Vadivasova ◽  
Andrey S. Kopeikin ◽  
Jürgen Kurths ◽  
Galina I. Strelkova
Keyword(s):  
Probability Measure ◽  
Invariant Probability Measure ◽  
Chaotic Attractors
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