scholarly journals Exponents, attractors and Hopf decompositions for interval maps

1990 ◽  
Vol 10 (4) ◽  
pp. 717-744 ◽  
Author(s):  
Gerhard Keller
Keyword(s):  
Schwarzian Derivative ◽  
Finite Union ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Closed Intervals ◽  
Absolutely Continuous Invariant Measure ◽  
Stable Periodic ◽  
Uniform Hyperbolicity ◽  
Absolutely Continuous Invariant

AbstractOur main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following:(1) There is a set CT such that the ω-limit of Lebesgue-a.e. point equals CT. CT is a finite union of closed intervals or it coincides with the closure of the critical orbit.(2) There is a constant λT such that for Lebesgue-a.e. x.(3) λT > 0 if and only if T has an absolutely continuous invariant measure of positive entropy.(4) , i.e. uniform hyperbolicity on periodic points implies λT > 0, and λT < 0 implies the existence of a stable periodic orbit.

scholarly journals RETURN TIME STATISTICS OF INVARIANT MEASURES FOR INTERVAL MAPS WITH POSITIVE LYAPUNOV EXPONENT

2009 ◽  
Vol 09 (01) ◽  
pp. 81-100 ◽  
Author(s):  
HENK BRUIN ◽  
MIKE TODD
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measures ◽  
Return Time ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Absolutely Continuous Invariant Measure ◽  
Normal Fluctuations ◽  
Return Time Statistics ◽  
Absolutely Continuous Invariant

We prove that multimodal maps with an absolutely continuous invariant measure have exponential return time statistics around almost every point. We also show a "polynomial Gibbs property" for these systems, and that the convergence to the entropy in the Ornstein–Weiss formula has normal fluctuations. These results are also proved for equilibrium states of some Hölder potentials.

Fibonacci maps re(aℓ)visited

1995 ◽  
Vol 15 (1) ◽  
pp. 99-120 ◽  
Author(s):  
Gerhard Keller ◽  
Tomasz Nowicki
Keyword(s):  
Invariant Measure ◽  
Critical Point ◽  
Schwarzian Derivative ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractWe prove that unimodal Fibonacci maps with negative Schwarzian derivative and a critical point of order ℓ have a finite absolutely continuous invariant measure if ℓ ∈ (1 ℓ1) where ℓ1 is some number strictly greater than 2. This extends results of Lyubich and Milnor for the case ℓ = 2.

CHAOTIC BEHAVIOR OF HIGHER DIMENSIONAL TRANSFORMATIONS DEFINED ON COUNTABLE PARTITIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 1045-1049
Author(s):  
A. BOYARSKY ◽  
Y. S. LOU
Keyword(s):  
Invariant Measure ◽  
Additional Condition ◽  
Invariant Measures ◽  
Chaotic Behavior ◽  
Finite Collection ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Higher Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collection. An example is given.

CONVERGENCE OF INVARIANT MEASURES FOR DEGENERATING SEQUENCES OF HYPERBOLIC MAPPINGS WITH SINGULARITIES

1996 ◽  
Vol 06 (06) ◽  
pp. 1143-1151
Author(s):  
E. A. SATAEV
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant ◽  
Expanding Mapping

This paper is devoted to presenting and giving a sketch of the proof of the theorem which states that, if the sequence of hyperbolic mappings with singularities converges to degenerating piecewise expanding mapping, then the corresponding sequence of measures of a Sinai-Bowen-Ruelle type converges to an absolutely continuous invariant measure.

scholarly journals Optimal transport and dynamics of expanding circle maps acting on measures

2012 ◽  
Vol 33 (2) ◽  
pp. 529-548 ◽  
Author(s):  
BENOÎT KLOECKNER
Keyword(s):  
Invariant Measure ◽  
Optimal Transport ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Expanding Circle ◽  
Rigorous Framework ◽  
Infinite Multiplicity ◽  
Set Up ◽  
Absolutely Continuous Invariant

AbstractIn this paper we compute the derivative of the action on probability measures of an expanding circle map at its absolutely continuous invariant measure. The derivative is defined using optimal transport: we use the rigorous framework set up by Gigli to endow the space of measures with a kind of differential structure. It turns out that 1 is an eigenvalue of infinite multiplicity of this derivative, and we deduce that the absolutely continuous invariant measure can be deformed in many ways into atomless, nearly invariant measures. We also show that the action of standard self-covering maps on measures has positive metric mean dimension.

scholarly journals Estimating invariant measures and Lyapunov exponents

1996 ◽  
Vol 16 (4) ◽  
pp. 735-749 ◽  
Author(s):  
Brian R. Hunt
Keyword(s):  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Invariant Measures ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Recent Interest ◽  
Absolutely Continuous Invariant Measure ◽  
Time Averages ◽  
Absolutely Continuous Invariant

AbstractThis paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius—Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.

Sign in / Sign up

Export Citation Format

Share Document