On non-uniform hyperbolicity assumptions in one-dimensional dynamics

Li, Huaibin; Shen, Weixiao

doi:10.1007/s11425-010-3134-4

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A Priori Bounds and Backward Contraction1

Some Facts1

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Cited by 10 publications

(2 citation statements)

References 14 publications

(28 reference statements)

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“…In the following theorem we summarize and complement results in [8,9], when restricted to unicritical maps. We state it in a stronger form than what is needed for this section.…”

Section: A Priori Bounds and Backward Contractionmentioning

confidence: 71%

On Poincaré Series of Unicritical Polynomials at the Critical Point

Rivera-Letelier

Shen²

2013

Commun. Math. Stat.

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“…In the following theorem we summarize and complement results in [8,9], when restricted to unicritical maps. We state it in a stronger form than what is needed for this section.…”

Section: A Priori Bounds and Backward Contractionmentioning

confidence: 71%

On Poincaré Series of Unicritical Polynomials at the Critical Point

Rivera-Letelier

Shen²

2013

Commun. Math. Stat.

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“…The notion of 'backward contraction' was introduced in [32]. Actually, the converse of the proposition is also true, see [24]. We shall use the following consequence of the proposition.…”

Section: Some Factsmentioning

confidence: 96%

On stochastic stability of non-uniformly expanding interval maps

Shen

2013

Proceedings of the London Mathematical Society

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Nice Inducing Schemes and the Thermodynamics of Rational Maps

Przytycki

Rivera-Letelier

2010

Commun. Math. Phys.

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Abstract:We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential −t ln f , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in R, with at most two exceptions.

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On non-uniform hyperbolicity assumptions in one-dimensional dynamics

Cited by 10 publications

References 14 publications

On Poincaré Series of Unicritical Polynomials at the Critical Point

On Poincaré Series of Unicritical Polynomials at the Critical Point

On stochastic stability of non-uniformly expanding interval maps

Nice Inducing Schemes and the Thermodynamics of Rational Maps

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