scholarly journals Induced maps, Markov extensions and invariant measures in one-dimensional dynamics

1995 ◽  
Vol 168 (3) ◽  
pp. 571-580 ◽  
Author(s):  
H. Bruin
Keyword(s):  
Invariant Measures ◽  
One Dimensional ◽  
Induced Maps ◽  
Markov Extensions

scholarly journals Lifting measures to inducing schemes

2008 ◽  
Vol 28 (2) ◽  
pp. 553-574 ◽  
Author(s):  
YA. B. PESIN ◽  
S. SENTI ◽  
K. ZHANG
Keyword(s):  
Metric Spaces ◽  
Borel Probability Measure ◽  
Equilibrium Measures ◽  
One Dimensional ◽  
Compact Metric Spaces ◽  
Borel Probability ◽  
One Dimensional Maps ◽  
Induced Maps ◽  
Inducing Schemes ◽  
Markov Extensions

AbstractIn this paper we study the liftability property for piecewise continuous maps of compact metric spaces, which admit inducing schemes in the sense of Pesin and Senti [Y. Pesin and S. Senti. Thermodynamical formalism associated with inducing schemes for one-dimensional maps. Mosc. Math. J.5(3) (2005), 669–678; Y. Pesin and S. Senti. Equilibrium measures for maps with inducing schemes. Preprint, 2007]. We show that under some natural assumptions on the inducing schemes—which hold for many known examples—any invariant ergodic Borel probability measure of sufficiently large entropy can be lifted to the tower associated with the inducing scheme. The argument uses the construction of connected Markov extensions due to Buzzi [J. Buzzi. Markov extensions for multi-dimensional dynamical systems. Israel J. Math.112 (1999), 357–380], his results on the liftability of measures of large entropy, and a generalization of some results by Bruin [H. Bruin. Induced maps, Markov extensions and invariant measures in one-dimensional dynamics. Comm. Math. Phys.168(3) (1995), 571–580] on relations between inducing schemes and Markov extensions. We apply our results to study the liftability problem for one-dimensional cusp maps (in particular, unimodal and multi-modal maps) and for some multi-dimensional maps.

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.

Higher Dimensional Asymptotic Cycles

2003 ◽  
Vol 55 (3) ◽  
pp. 636-648 ◽  
Author(s):  
Sol Schwartzman
Keyword(s):  
Invariant Measure ◽  
Compact Manifold ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Stationary Points ◽  
Dimensional Case ◽  
One Dimensional ◽  
Smooth Flow ◽  
Higher Dimensional ◽  
The One

AbstractGiven a p-dimensional oriented foliation of an n-dimensional compact manifold Mn and a transversal invariant measure τ, Sullivan has defined an element of Hp(Mn; R). This generalized the notion of a μ-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure μ. In this one-dimensional case there was a natural 1—1 correspondence between transversal invariant measures τ and invariant measures μ when one had a smooth flow without stationary points.For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

scholarly journals Iteration of piecewise linear maps on an interval

1981 ◽  
Vol 24 (3) ◽  
pp. 433-451 ◽  
Author(s):  
James B. McGuire ◽  
Colin J. Thompson
Keyword(s):  
Invariant Measures ◽  
Piecewise Linear ◽  
Point Of View ◽  
Complete Analysis ◽  
Functional Composition ◽  
One Dimensional ◽  
Linear Maps ◽  
Piecewise Linear Maps ◽  
Parameter Values ◽  
Dense Set

A complete analysis is given of the iterative properties of two piece-piecewise linear maps on an interval, from the point of view of a doubling transformation obtained by functional composition and rescaling. We show how invariant measures may be constructed for such maps and that parameter values where this may be done form a dense set in a one-dimensional subset of parameter space.

Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points

2000 ◽  
Vol 20 (5) ◽  
pp. 1519-1549 ◽  
Author(s):  
ROLAND ZWEIMÜLLER
Keyword(s):  
Limit Theorem ◽  
Fixed Points ◽  
Limit Theorems ◽  
Invariant Measures ◽  
Transfer Operator ◽  
Large Collection ◽  
Interval Maps ◽  
Infinite Measure ◽  
Ergodic Properties ◽  
Markov Extensions

We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do not assume the existence of a Markov partition but only require the first image of the fundamental partition to be finite, we use canonical Markov extensions to first prove pointwise dual-ergodicity, which, together with an identification of wandering rates, leads to distributional limit theorems. We show that $T$ satisfies Rohlin's formula and prove a variant of the Shannon–McMillan–Breiman theorem. Moreover, we give a stronger limit theorem for the transfer operator providing us with a large collection of uniform and Darling–Kac sets. This enables us to apply recent results from fluctuation theory.

scholarly journals Invariant measures for interval translations and some other piecewise continuous maps

2020 ◽  
Vol 15 ◽  
pp. 15
Author(s):  
Sergey Kryzhevich
Keyword(s):  
Invariant Measure ◽  
Compact Manifold ◽  
Invariant Measures ◽  
Interval Exchange ◽  
One Dimensional ◽  
Simple Result ◽  
Continuous Maps ◽  
Piecewise Continuous ◽  
Smooth Partition ◽  
Borel Probability

We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps) a Borel probability non-atomic invariant measure exists for any map. We use this result to demonstrate that any interval translation map endowed with such a measure is metrically equivalent to an interval exchange map. Finally, we study the general case of piecewise continuous maps and prove a simple result on existence of an invariant measure provided all discontinuity points are wandering.

scholarly journals Transitivity and invariant measures for the geometric model of the Lorenz equations

1984 ◽  
Vol 4 (4) ◽  
pp. 605-611 ◽  
Author(s):  
Clark Robinson
Keyword(s):  
Lebesgue Measure ◽  
Invariant Measures ◽  
Geometric Model ◽  
Ergodic Measure ◽  
Lorenz Equations ◽  
Poincaré Maps ◽  
End Points ◽  
One Dimensional ◽  
Topologically Transitive ◽  
The One

AbstractThis paper concerns perturbations of the geometric model of the Lorenz equations and their associated one-dimensional Poincaré maps. R. Williams has shown that if a map of the interval of the type arising from the Lorenz equations satisfies f′(x) > 2½ everywhere except at the discontinuity then f is locally eventually onto (l.e.o.) and so topologically transitive [14]. We show roughly that if are the end points of the interval, and their iterates stay on the same side of the point of discontinuity for 0 ≤ j ≤ k, and f′(x) > 21/(k+1) everywhere, then f is l.e.o. Secondly, we show that the one-dimensional Poincaré map of any Cr perturbation of the geometric model (for large enough r) has an ergodic measure which is equivalent to Lebesgue measure. This result follows by showing it is C1+α and satisfies a theorem of Keller, Wong, Lasota, Li & Yorke.

scholarly journals Invariant measures for some one-dimensional attractors

1982 ◽  
Vol 2 (3-4) ◽  
pp. 317-337 ◽  
Author(s):  
M. V. Jacobson
Keyword(s):  
Invariant Measures ◽  
Small Perturbations ◽  
One Dimensional ◽  
Quadratic Maps ◽  
Hyperbolic Attractors

AbstractWe consider certain non-invertible maps of the square which are extensions of the quadratic maps of the interval and their small perturbations. We show that several maps of the type possess attractors which are not hyperbolic but have invariant measures similar to Bowen-Ruelle measures for hyperbolic attractors.

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