Invariant measures exist under a summability condition for unimodal maps

1991 ◽  
Vol 105 (1) ◽  
pp. 123-136 ◽  
Author(s):  
Tomasz Nowicki ◽  
Sebastian van Strien
Keyword(s):  
Invariant Measures ◽  
Unimodal Maps ◽  
Summability Condition

The existence of absolutely continuous invariant measures is not a topological invariant for unimodal maps

1998 ◽  
Vol 18 (3) ◽  
pp. 555-565 ◽  
Author(s):  
HENK BRUIN
Keyword(s):  
Probability Measure ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Topological Invariant ◽  
Absolutely Continuous ◽  
Unimodal Maps ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant ◽  
Critical Order

Within the class of S-unimodal maps with fixed critical order it is shown that the existence of an absolutely continuous invariant probability measure is not a topological invariant.

Metric properties of non-renormalizableS-unimodal maps. Part I: Induced expansion and invariant measures

1994 ◽  
Vol 14 (4) ◽  
pp. 721-755 ◽  
Author(s):  
Michael Jakobson ◽  
Grzegorz Światek
Keyword(s):  
Critical Point ◽  
Schwarzian Derivative ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Limit Set ◽  
Countable Union ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Starting Condition ◽  
Metric Properties

AbstractFor an arbitrary non-renormalizable unimodal map of the interval,f:I→I, with negative Schwarzian derivative, we construct a related mapFdefined on a countable union of intervals Δ. For each interval Δ,Frestricted to Δ is a diffeomorphism which coincides with some iterate offand whose range is a fixed subinterval ofI. IfFsatisfies conditions of the Folklore Theorem, we callfexpansion inducing. Letcbe a critical point off. Forfsatisfyingf″(c) ≠ 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds forf: the ω-limit set of Lebesgue almost every point is the interval [f2,f(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.

Topological conditions for the existence of invariant measures for unimodal maps

1994 ◽  
Vol 14 (3) ◽  
pp. 433-451 ◽  
Author(s):  
H. Bruin
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Invariant Measures ◽  
Finite Measure ◽  
Critical Value ◽  
Absolutely Continuous ◽  
Unimodal Maps ◽  
Derivatives Of

AbstractWe present a class of S-unimodal maps having an invariant measure which is absolutely continuous with respect to Lebesgue measure. This measure can often be proved to be finite. We give an example of a map which has such a finite measure and for which the lim inf of the derivatives of the iterates of the map in the critical value is finite. It will be shown that all topologically conjugate non-flat S-unimodal maps have the same properties.

SMOOTH SYMMETRIC AND LORENZ MODELS FOR UNIMODAL MAPS

2003 ◽  
Vol 13 (11) ◽  
pp. 3353-3371 ◽  
Author(s):  
MING-CHIA LI ◽  
MIKHAIL MALKIN
Keyword(s):  
Turning Point ◽  
Invariant Measures ◽  
Structural Instability ◽  
Symmetric Model ◽  
Lorenz Model ◽  
Unimodal Maps ◽  
Unimodal Map ◽  
Ergodic Properties ◽  
Discontinuous Map ◽  
Higher Derivatives

For a given unimodal map F:I→I on the interval I, we consider symmetric unimodal maps (models) so that they are conjugate to F. The question motivated by [Gambaudo & Tresser, 1992] is the following: whether it is possible for symmetric model to preserve smoothness of the initial map F? We construct a symmetric model which is proved to be as smooth as F provided F has a nonflat turning point with sufficient "reserve of local evenness" at the turning point (in terms of one-sided higher derivatives at the turning point, see Definition 2.4 and Theorem 2.7). We also consider from different points of view the relationship between dynamical and ergodic properties of unimodal maps and of symmetric Lorenz maps. In particular, we present a one-to-one correspondence preserving the measure theoretic entropy, between the set of invariant measures of a symmetric unimodal map F and the set of symmetric invariant measures of the Lorenz model of F (Theorem 3.5), where by Lorenz model of F we mean the discontinuous map obtained from F by reversing its decreasing branch. Finally we extend for nonsymmetric unimodal maps, the result of Gambaudo and Tresser [1992] on Ck structural instability of the maps whose rotation interval has irrational end point (answering a question from [Gambaudo & Tresser, 1992]).

Extending Quasi-Invariant Measures by Using Subgroups of a Given Group

2003 ◽  
Vol 10 (2) ◽  
pp. 247-255
Author(s):  
A. Kharazishvili
Keyword(s):  
Invariant Measures ◽  
Uncountable Group

Abstract A method of extending σ-finite quasi-invariant measures given on an uncountable group, by using a certain family of its subgroups, is investigated.

scholarly journals Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 80
Author(s):  
Sergey Kryzhevich ◽  
Viktor Avrutin ◽  
Nikita Begun ◽  
Dmitrii Rachinskii ◽  
Khosro Tajbakhsh
Keyword(s):  
Risk Management ◽  
Invariant Measure ◽  
Invariant Measures ◽  
Interval Exchange ◽  
Management Model ◽  
Closing Lemma ◽  
Symbolic Model ◽  
Metric Properties ◽  
Ergodic Measures ◽  
Maximal Invariant

We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and demonstrated that an ITM, endowed with such a measure, is metrically conjugated to an interval exchange map (IEM). This allowed us to extend some properties of IEMs (e.g., an estimate of the number of ergodic measures and the minimality of the symbolic model) to ITMs. Further, we proved a version of the closing lemma and studied how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.

Sign in / Sign up

Export Citation Format

Share Document