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Yunping Jiang

Publications and source records attributed to Yunping Jiang.

At least 19 recordsLinked to original sources
arXiv2026

Space of Geometric Gibbs Measures: Uniqueness, Ergodicity, and Teichmüller Complex Manifold Structure

Building on our previous work, in which geometric Gibbs measures were introduced to complete the space of classical Gibbs measures under the Teichmüller metric, we establish the uniqueness of geometric Gibbs measures in this paper. This result completes the construction of a Teichmüller complex manifold structure on the space of all geometric Gibbs measures. Using the welding technique, we uncover a connection between geometric Gibbs measures and harmonic measures on dynamical quasicircles. This connection allows us to establish asymptotic conformal rigidity for these dynamical quasicircles. Finally, we establish a foundation for the systematic study of the ergodic properties of geometric Gibbs measures.

math.DS↗
arXiv2026

Time-Averages along Partial Orbits for Dynamical Systems

We investigate the convergence of time-averages of continuous functions along partial orbits for continuous dynamical systems on compact metric spaces. Unlike the classical focus of ergodic theory on convergence almost everywhere, our study emphasizes convergence at every point, motivated by problems in number theory. To this end, we consider a sequence having a uniform behavior property in the natural numbers. We also define and study two structural properties of dynamical systems along such partial orbits: minimal mean Lyapunov stability and minimal mean attractability. Our main result shows that if a continuous dynamical system is both minimally mean Lyapunov stable and minimally mean attractable along a given uniformly behaved sequence, and is minimally uniquely ergodic, then the time-average of any continuous function along the corresponding partial orbit converges to the space average at every point.

math.NT↗
arXiv2026

Typical periodic optimization and freezing phase transition: Dini potentials and beyond

We investigate the Typical Periodic Optimization (TPO) Conjecture and freezing phase transitions for Dini potentials and beyond. We show that the TPO conjecture fails already at Dini regularity. We identify a sharp regularity threshold for freezing phase transitions, which we call weak log-log Dini. Both results reveal the stability of aperiodic ground states. Moreover, we identify Dini regularity as the sharp threshold for a quantitatitve analog of the freezing phase transition.

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arXiv2026

Transversality at Virtual Centers

We consider the family of transcendental meromorphic functions with finitely many singular values, none of whose asymptotic values is infinity. We prove that transversality holds at any boundary point of a shell component in a dynamically natural slice of the parameter space at which an asymptotic value is a prepole.

math.CV↗
arXiv2026

Geometry in the Furstenberg Conjecture

We explore the geometric aspects of the Furstenberg conjecture, proving that a non-atomic probability measure on the unit circle, invariant under both $p$- and $q$-actions for coprime integers $p,q>1$, must be the Lebesgue measure if it exhibits balanced geometry for one of these actions. Within rigidity theory, we show that balanced geometry is equivalent to the Lipschitz property. A consequence is that an orientation-preserving homeomorphism of the circle conjugating both $p$- and $q$-actions and preserving the Lebesgue measure must be the identity if one of these conjugations satisfies the Lipschitz property. Our approach does not rely solely on ergodicity, and we conclude by proposing conjectures and open problems that frame the Furstenberg conjecture through geometric and quasisymmetric perspectives.

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arXiv2026

Meromorphic functions whose action on their Julia sets is Non-Ergodic

Nevanlinna functions are meromorphic functions with a finite number of asymptotic values and no critical values. In [KK2] it was proved that if the orbits of all the asymptotic values accumulate on a compact set on which the function acts as a repeller, then the function acts ergodically on its Julia set. In [CJK4] we proved the action of the function on its Julia set is still ergodic if some, but not all of the asymptotic values land on infinity, and the remaining ones land on a compact repeller. In this paper, we complete the characterization of ergodicity for Nevanlinna functions by proving that if all the asymptotic values land on infinity, then the Julia set is the whole sphere and the action of the map there is non-ergodic.

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arXiv2026

Ruelle's zeta function for non-Archimedean rational maps

We studied the transfer operators defined over $\mathbb{C}_p$-valued analytic functions for subhyperbolic rational maps on $\mathbb{Q}_p$, and showed that the corresponding Ruelle's zeta functions are meromorphic on $\mathbb{C}_p$. We also used $\mathbb{R}$-valued transfer operators to study the shape of the corresponding Julia sets, and proved a Levin-Sodin-Yuditski type identity for general rational maps on $\mathbb{C}_p$. In all the results above, $\mathbb{Q}_p$ can be replaced with any non-Archimedean local field with characteristic $0$, and $\mathbb{C}_p$ the metric completion of its algebraic closure.

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arXiv2025

Orders of Oscillation Motivated by Sarnak's Conjecture--Part II

This work is a continuation of [13]. We study the linear disjointness between higher-order oscillating sequences and nonlinear dynamical systems. Specifically, we prove that any oscillating sequence of order $m=d+k-1$ and any simple polynomial skew product of degree $k$ on the $d$-Euclidean space are linearly disjoint. Additionally, we demonstrate that any oscillating sequence of order $d$ and any minimal mean attractable and minimal quasi-discrete spectrum dynamical system of order $d$ are linearly disjoint. Finally, we introduce multi-linearly disjoint sequences and construct examples of such sequences.

math.DS↗
arXiv2024

Ergodicity in some families of Nevanlinna Functions

We study Nevanlinna functions f that are transcendental meromorphic functions having N asymptotic values and no critical values. In [KK] it was proved that if the orbits of all the asymptotic values have accumulation sets that are compact and on which f is a repeller, then f acts ergodically on its Julia set. In this paper, we prove that if some, but not all of the asymptotic values have this property, while the others are prepoles, the same holds true. This is the first paper to consider this mixed case.

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arXiv2023

Meromorphic functions whose action on their Julia sets is non-ergodic

We study transcendental meromorphic functions having two prepole asymptotic values and no critical values. We prove that these functions acting on their Julia sets are non-ergodic, which illustrates the antithesis of the Keen-Kotus result in [KK2] on the ergodicity of another subfamily of functions with two asymptotic values and no critical values.

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arXiv2021

Slices of Parameter Space for Meromorphic Maps with Two Asymptotic Values

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work in [GK] on degree 2 rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further, is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted.

math.CV↗
arXiv2021

Symmetric Rigidity for Circle Endomorphisms with Bounded Geometry

Let $f$ and $g$ be two circle endomorphisms of degree $d\geq 2$ such that each has bounded geometry, preserves the Lebesgue measure, and fixes $1$. Let $h$ fixing $1$ be the topological conjugacy from $f$ to $g$. That is, $h\circ f=g\circ h$. We prove that $h$ is a symmetric circle homeomorphism if and only if $h=Id$. Many other rigidity results in circle dynamics follow from this very general symmetric rigidity result.

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arXiv2020

Accessible Boundary Points in the Shift Locus of a Familiy of Meromorphic Functions with Two Finite Asymptotic Values

In this paper we continue the study, began in \cite{CJK2}, of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values and an attracting fixed point. If we fix the multiplier of the fixed point, either of the two asymptotic values determines a one-dimensional parameter slice for this family. We proved that the bifurcation locus divides this parameter slice into three regions, two of them analogous to the Mandelbrot set and one, the shift locus, analogous to the complement of the Mandelbrot set. In \cite{FK, CK} it was proved that the points in the bifurcation locus corresponding to functions with a parabolic cycle, or those for which some iterate of one of the asymptotic values lands on a pole are accessible boundary points of the hyperbolic components of the Mandelbrot-like sets. Here we prove these points, as well as the points where some iterate of the asymptotic value lands on a repelling periodic cycle, are also accessible from the shift locus.

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arXiv2018

Cycle Doubling, Merging And Renormalization in the Tangent Family

In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $\{ T_t(z)=i t\tan z\}_{0< t\leq π}$. Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as $t$ increases we find single instances of "period quadrupling", "period splitting" and standard "period doubling"; there follows a general pattern of "period merging" where two attracting cycles of period $2^n$ "merge" into one attracting cycle of period $2^{n+1}$, and "cycle doubling" where an attracting cycle of period $2^{n+1}$ "becomes" two attracting cycles of the same period. We use renormalization to prove the existence of these bifurcation parameters. The uniqueness of the cycle doubling and cycle merging parameters is quite subtle and requires a new approach. To prove the cycle doubling and merging parameters are, indeed, unique, we apply the concept of "holomorphic motions" to our context. In addition, we prove that there is an "infinitely renormalizable" tangent map $T_{t_\infty}$. It has no attracting or parabolic cycles. Instead, it has a strange attractor contained in the real and imaginary axes which is forward invariant and minimal under $T^2_{t_\infty}$. The intersection of this strange attractor with the real line consists of two binary Cantor sets and the intersection with the imaginary line is totally disconnected, perfect and unbounded.

math.DS↗
arXiv2018

Monodromy, liftings of holomorphic maps, and extensions of holomorphic motions

We study monodromy of holomorphic motions and show the equivalence of triviality of monodromy of holomorphic motions and extensions of holomorphic motions to continuous motions of the Riemann sphere. We also study liftings of holomorphic maps into certain Teichmüller spaces. We use this "lifting property" to prove that, under the condition of trivial monodromy, any holomorphic motion of a closed set in the Riemann sphere, over a hyperbolic Riemann surface, can be extended to a holomorphic motion of the sphere, over the same parameter space. We conclude that this extension can be done in a conformally natural way.

math.CV↗
arXiv2017

Zero Entropy Interval Maps And MMLS-MMA Property

We prove that the flow generated by any interval map with zero topological entropy is minimally mean-attractable (MMA) and minimally mean-L-stable (MMLS). One of the consequences is that any oscillating sequence is linearly disjoint with all flows generated by interval maps with zero topological entropy. In particular, the Möbius function is orthogonal to all flows generated by interval maps with zero topological entropy (Sarnak's conjecture for interval maps). Another consequence is a non-trivial example of a flow having the discrete spectrum.

math.DS↗
arXiv2016

Higher Order Oscillating Sequences, Affine Distal Flows on the $d$-Torus, and Sarnak's Conjecture

In this paper, we give two precise definitions of a higher order oscillating sequence and show the importance of this concept in the study of Sarnak's conjecture. We prove that any higher order oscillating sequence of order $d$ is linearly disjoint from all affine distal flows on the $d$-torus for all $d\geq 2$. One consequence of this result is that any higher order oscillating sequence of order $2$ is linearly disjoint from all affine flows on the $2$-torus with zero topological entropy. In particular, this reconfirms Sarnak's conjecture for all affine flows on the $2$-torus with zero topological entropy and for all affine distal flows on the $d$-torus for all $d\geq 2$.

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