Prime number theorems and holonomies for hyperbolic rational maps

Oh, Hee; Winter, Dale

doi:10.1007/s00222-016-0693-1

Cited by 16 publications

(39 citation statements)

References 18 publications

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“…For lattices, Theorem 1.3 was obtained by Sarnak and Wakayama [SW99] using the Selberg trace formula. We also remark that the analogue of Theorem 1.3 for hyperbolic rational maps on the Riemann sphere was obtained by Oh and Winter [OW17] and hence adding to Sullivan's dictionary: holonomies are exponentially equidistributed both for primitive periodic orbits of hyperbolic rational maps on the Riemann sphere and primitive closed geodesics in convex cocompact hyperbolic manifolds.…”

Section: Introductionmentioning

confidence: 72%

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Sarkar¹,

Winter²

2021

Compositio Math.

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The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

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Section: Introductionmentioning

confidence: 72%

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Sarkar¹,

Winter²

2021

Compositio Math.

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“…This was later used by Pollicott and Sharp to obtain an analogue of the prime number theorem with an exponential error term for closed geodesics on a compact negatively curved surface [14]. Naud adapted Dolgopyat's analysis to prove a similar result for convex co-compact surfaces [7] as well as Oh and Winter whose work was in the current setting of expanding rational maps [8]. We use a similar approach to obtain bounds on the spectral radii of a family of transfer operators in order to extract our asymptotic result in the final section.…”

Section: Decay Estimatesmentioning

confidence: 99%

“…We therefore have all the properties required to get the following theorem. Theorem 3.7 (Theorem 2.7, [8]). Suppose that the Julia set of f is not contained in a circle in C. Then there exist C > 0 and ρ ∈ (0, 1) such that for any w ∈ C 1 (U ) with w (|b|+|k|) ≤ 1 and any n ∈ N…”

Section: Decay Estimatesmentioning

confidence: 99%

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Statistics of multipliers for hyperbolic rational maps

Sharp¹,

Stylianou²

2022

DCDS

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<p style='text-indent:20px;'>In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.</p>

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“…It arises naturally in several important problems, like in the construction of physical measures, as in the pioneering work of Sinaȋ [Sin72], Ruelle [Rue76], and Bowen [Bow75]. The pressure of the geometric potential is connected, among other things, to several multifractal spectra, and large deviations rate functions, see for example [BMS03, Lemma 2], [CRT16,GPR10,GPR16], [KN92, Theorems 1.2 and 1.3], [PRL11, Appendix B], [OW17] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Sensitive Dependence of Geometric Gibbs States at Positive Temperature

Coronel

Rivera-Letelier

2019

Commun. Math. Phys.

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We give the first example of a smooth family of real and complex maps having sensitive dependence of geometric Gibbs states at positive temperature. This family consists of quadratic-like maps that are non-uniformly hyperbolic in a strong sense. We show that for a dense set of maps in the family the geometric Gibbs states do not converge at positive temperature. These are the first examples of non-convergence at positive temperature in statistical mechanics or the thermodynamic formalism, and answers a question of van Enter and Ruszel. We also show that this phenomenon is robust: There is an open set of analytic 2-parameter families of quadratic-like maps that exhibit sensitive dependence of geometric Gibbs states at positive temperature. * In §3.1 we extend the definition of essentially topologically exact map to certain quadratic-like maps that are not necessarily real.On the other hand, for t in R/Z the setis called the external ray of angle t of M. We say that R(t) lands at a point z in C, if Φ −1 (r exp(2πit)) converges to z as r ց 1. When this happens z belongs to ∂M.Note that for a real parameter c, every puzzle piece intersecting the real line is invariant under complex conjugation. Since puzzle pieces are simply-connected, it follows that the intersection of such a puzzle piece with R is an interval.Definition 2.2 (Yoccoz para-puzzle † ). Given an integer n ≥ 0, put J n := {t ∈ [1/3, 2/3] | 2 n t (mod 1) ∈ {1/3, 2/3}}, let X n be the intersection of W with the open region in the parameter plane bounded by the equipotential E(2 −n ) of M, and put I n := ∂X n ∪ X n ∩ t∈Jn cl(R(t)) .Then the Yoccoz para-puzzle of W is the sequence of graphs (I n ) +∞ n=0 . The parapuzzle pieces of depth n are the connected components of X n \ I n . The para-puzzle piece of depth n containing a parameter c is denoted by P n (c). † In contrast to [Roe00], we only consider the para-puzzle in the wake W.

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Prime number theorems and holonomies for hyperbolic rational maps

Cited by 16 publications

References 18 publications

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

Statistics of multipliers for hyperbolic rational maps

Sensitive Dependence of Geometric Gibbs States at Positive Temperature

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