On the relations between positive Lyapunov exponents, positive
  entropy, and sensitivity for interval maps

Blaya, Alejo Barrio; López, Vı́ctor Jiménez

doi:10.3934/dcds.2012.32.433

Cited by 11 publications

(2 citation statements)

References 20 publications

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“…Here and elsewhere A ≡ B means "A is shorthand for B" 4. This definition agrees with the multiplicative approximation framework considered in[15], but not the one considered in[14]; see comments after Proposition 4.1 for more details.…”

supporting

confidence: 68%

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

Das

Fishman

Simmons

et al. 2017

Sel. Math. New Ser.

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We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis ('98) resolving Sprindžuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss ('04), hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, of the Patterson-Sullivan measures of all nonplanar geometrically finite groups, and of the Gibbs measures (including conformal measures) of infinite iterated function systems. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW's sufficient conditions for extremality. In Part I, we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, thus proving a conjecture of KLW. We also prove the "inherited exponent of irrationality" version of this theorem, describing the relationship between the Diophantine properties of certain subspaces of the space of matrices and measures supported on these subspaces. In subsequent papers, we exhibit numerous examples of quasi-decaying measures, in support of the thesis that "almost any measure from dynamics and/or fractal geometry is quasi-decaying". We also discuss examples of non-extremal measures coming from dynamics, illustrating where the theory must halt.

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supporting

confidence: 68%

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

Das

Fishman

Simmons

et al. 2017

Sel. Math. New Ser.

View full text Add to dashboard Cite

show abstract

“…We show that ϕ • π ∈ A(Σ β , {0}), and it immediately follows that tϕ • π ∈ A(Σ β , {0}) for all t. For t < 1, we then verify the hypotheses of Theorem 3.12 to establish uniqueness of equilibrium states. The second and third parts of the theorem follow directly from the inequality h µ (f ) ≤ log |f ′ (x)| dµ for all ergodic f -invariant measures, which holds for this family of maps by [Hof91, Theorem 1] (see also [BBJL12,Theorem 7.1]). Let x 0 be the solution of x + γx 1+ε = 1, and let x n be the unique number with 0 < x n < x 0 and x n−1 = f (x n ).…”

Section: 3mentioning

confidence: 99%

Equilibrium states beyond specification and the Bowen property

Climenhaga

Thompson

2012

Journal of the London Mathematical Society

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It is well known that for expansive maps and continuous potential functions, the specification property (for the map) and the Bowen property (for the potential) together imply the existence of a unique equilibrium state. We consider symbolic spaces that may not have specification and potentials that may not have the Bowen property, and give conditions under which uniqueness of the equilibrium state can still be deduced. Our approach is to ask that the collection of cylinders which are obstructions to the specification property or the Bowen property is small in an appropriate quantitative sense. This allows us to construct an ergodic equilibrium state with a weak Gibbs property, which we then use to prove uniqueness. We do not use inducing schemes or the Perron–Frobenius operator, and we strengthen some previous results obtained using these approaches. In particular, we consider β‐shifts and show that the class of potential functions with unique equilibrium states strictly contains the set of potentials with the Bowen property. We give applications to piecewise monotonic interval maps, including the family of geometric potentials, for example, which have both indifferent fixed points and a non‐Markov structure.

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Destabilization and chaos induced by harvesting: insights from one-dimensional discrete-time models

López

Liz

2021

J. Math. Biol.

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On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps

Cited by 11 publications

References 20 publications

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures

Equilibrium states beyond specification and the Bowen property

Destabilization and chaos induced by harvesting: insights from one-dimensional discrete-time models

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