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.

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.

scholarly journals STATISTICAL PROPERTIES OF ONE-DIMENSIONAL MAPS WITH CRITICAL POINTS AND SINGULARITIES

2006 ◽  
Vol 06 (04) ◽  
pp. 423-458 ◽  
Author(s):  
K. DÍAZ-ORDAZ ◽  
M. P. HOLLAND ◽  
S. LUZZATTO
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Invariant Probability Measure ◽  
Return Time ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Time Asymptotics ◽  
Exponential Decay Of Correlations ◽  
One Dimensional Maps ◽  
Absolutely Continuous Invariant

We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for Hölder observations.

Piecewise smooth interval maps with non-vanishing derivative

2000 ◽  
Vol 20 (3) ◽  
pp. 749-773 ◽  
Author(s):  
ALE JAN HOMBURG
Keyword(s):  
Probability Measure ◽  
Periodic Orbits ◽  
Invariant Probability Measure ◽  
Piecewise Smooth ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Periodic Attractors ◽  
Absolutely Continuous Invariant

We consider the dynamics of piecewise smooth interval maps $f$ with a nowhere vanishing derivative. We show that if $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If, in addition all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelling, then some iterate of $f$ is expanding. In this case, $f$ admits an absolutely continuous invariant probability measure.

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.

scholarly journals Ergodic optimization in dynamical systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2593-2618 ◽  
Author(s):  
OLIVER JENKINSON
Keyword(s):  
Probability Measure ◽  
Invariant Measures ◽  
Model Problem ◽  
Thermodynamic Formalism ◽  
Invariant Probability Measure ◽  
Temperature Limit ◽  
Zero Temperature ◽  
Ergodic Optimization ◽  
Maximizing Measure ◽  
Time Average

Ergodic optimization is the study of problems relating to maximizing orbits and invariant measures, and maximum ergodic averages. An orbit of a dynamical system is called$f$-maximizing if the time average of the real-valued function$f$along the orbit is larger than along all other orbits, and an invariant probability measure is called$f$-maximizing if it gives$f$a larger space average than any other invariant probability measure. In this paper, we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.

scholarly journals Un exemple de transformation dilatante et C1 par morceaux de l'intervalle, sans probabilité absolument continue invariante

1989 ◽  
Vol 9 (1) ◽  
pp. 101-113 ◽  
Author(s):  
P. Gora ◽  
B. Schmitt
Keyword(s):  
Probability Measure ◽  
Modulus Of Continuity ◽  
Invariant Probability Measure ◽  
Negative Answer ◽  
Continuous Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Measure ◽  
Absolutely Continuous Invariant

AbstractWe construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..

scholarly journals Measure rigidity for algebraic bipermutative cellular automata

2007 ◽  
Vol 27 (6) ◽  
pp. 1965-1990 ◽  
Author(s):  
MATHIEU SABLIK
Keyword(s):  
Cellular Automata ◽  
Probability Measure ◽  
Cellular Automaton ◽  
Compact Group ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Invariant Subgroup ◽  
Shift Map ◽  
Bernoulli Measure ◽  
Measure Rigidity

AbstractLet $({\mathcal {A}^{\mathbb {Z}}} ,F)$ be a bipermutative algebraic cellular automaton. We present conditions that force a probability measure, which is invariant for the $ {\mathbb {N}} \times {\mathbb {Z}} $-action of F and the shift map σ, to be the Haar measure on Σ, a closed shift-invariant subgroup of the abelian compact group $ {\mathcal {A}^{\mathbb {Z}}} $. This generalizes simultaneously results of Host et al (B. Host, A. Maass and S. Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9(6) (2003), 1423–1446) and Pivato (M. Pivato. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12(4) (2005), 723–736). This result is applied to give conditions which also force an (F,σ)-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible affine expansive cellular automaton on $ {\mathcal {A}^{\mathbb {N}}} $.

scholarly journals Invariant measures for interval maps without Lyapunov exponents

2021 ◽  
pp. 1-29
Author(s):  
JORGE OLIVARES-VINALES
Keyword(s):  
Lyapunov Exponent ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Critical Points ◽  
Invariant Measures ◽  
Full Measure ◽  
Interval Maps ◽  
Interval Map

Abstract We construct an invariant measure for a piecewise analytic interval map whose Lyapunov exponent is not defined. Moreover, for a set of full measure, the pointwise Lyapunov exponent is not defined. This map has a Lorenz-like singularity and non-flat critical points.

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