arXiv subjects
Publications and source records attributed to Weixiao Shen.
We study the ergodic optimization problem over a real analytic expanding circle map. We show that in both the topological and the measure-theoretical senses, a typical $C^r$ performance function has a unique maximizing measure and the unique maximizing measure is supported on a periodic orbit, for $r=1,2,\dots,\infty,ω$.
Wandering Fatou components were recently constructed by Astorg et al for higher-dimensional holomorphic maps on projective spaces. Their examples are polynomial skew products with a parabolic invariant line. In this paper, we study this wandering domain problem for polynomial skew product $f$ with an attracting invariant line $L$ (which is the more common case). We show that if $f$ is unicritical (in the sense that the critical curve has a unique transversal intersection with $L$), then every Fatou component of $f$ in the basin of $L$ is an extension of a one-dimensional Fatou component of $f|_L$. As a corollary, there is no wandering Fatou component. We will also discuss the multicritical case under additional assumptions.
We prove that an asymmetric unimodal map has no wandering intervals.
For a real analytic periodic function $ϕ:\mathbb{R}\to\mathbb{R}^d$, an integer $b \ge 2$ and $λ\in(1/b,1)$, we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function $W(x)=\sum_{n=0}^{\infty}{λ^nϕ(b^nx)}$ are both equal to $$\min\left\{\log_{λ^{-1}}b,\,1+\left(\,d-q\,\right)\left(1+\log_bλ\right)\right\},$$ where $q = q(ϕ, b, λ)$ denotes the maximum dimension of all linear spaces $V < \mathbb{R}^d$ such that the projection $π_V W$ is Lipschitz.
In this paper we will give a short and elementary proof that critical relations unfold transversally in the space of rational maps.
In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in \cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcations at parabolic periodic points are generic, or one has persistency of periodic orbits with a fixed multiplier.
We consider the generalized Thue-Morse sequences $(t_n^{(c)})_{n\ge 0}$ ($c \in [0,1)$ being a parameter) defined by $t_n^{(c)} = e^{2πi c s_2(n)}$, where $s_2(n)$ is the sum of digits of the binary expansion of $n$. For the polynomials $σ_{N}^{(c)} (x) := \sum_{n=0}^{N-1} t_n^{(c)} e^{2πi n x}$, we have proved in [18] that the uniform norm $\|σ_N^{(c)}\|_\infty$ behaves like $N^{γ(c)}$ and the best exponent $γ(c)$ is computed. In this paper, we study the pointwise behavior and give a complete multifractal analysis of the limit $\lim_{n\to\infty}n^{-1}\log |σ_{2^n}^{(c)}(x)|$.
In this paper, we prove that for real analytic expanding circle maps, all optimizing measures of a real analytic potential function have zero entropy, unless the potential is cohomologous to constant. We use the group structure of the symbolic space to solve a transversality problem involved. We also discuss applications to optimizing measures for generic smooth potentials and to Lyapunov optimizing measures.
We introduce the notion of Bohr chaoticity, which is a topological invariant for topological dynamical systems, and which is opposite to the property required by Sarnak's conjecture. We prove the Bohr chaoticity for all systems which have a horseshoe and for all toral affine dynamical systems of positive entropy, some of which don't have a horseshoe. But uniquely ergodic dynamical systems are not Bohr chaotic.
Using quasiconformal surgery, we prove that any primitive, postcritically-finite hyperbolic polynomial can be tuned with an arbitrary generalized polynomial with non-escaping critical points, generalizing a result of Douady-Hubbard for quadratic polynomials to the case of higher degree polynomials. This solves affirmatively a conjecture by Inou and Kiwi on surjectivity of the renormalization operator on higher degree polynomials in one complex variable.
For a real analytic periodic function $ϕ:\mathbb{R}\to \mathbb{R}$, an integer $b\ge 2$ and $λ\in (1/b,1)$, we prove the following dichotomy for the Weierstrass-type function $W(x)=\sum\limits_{n\ge 0}{λ^nϕ(b^nx)}$: Either $W(x)$ is real analytic, or the Hausdorff dimension of its graph is equal to $2+\log_bλ$. Furthermore, given $b$ and $ϕ$, the former alternative only happens for finitely many $λ$ unless $ϕ$ is constant.
In this paper we will develop a general approach which shows that generalized "critical relations" of families of locally defined holomorphic maps on the complex plane unfold transversally. The main idea is to define a transfer operator, which is a local analogue of the Thurston pullback operator, using holomorphic motions. Assuming a so-called lifting property is satisfied, we obtain information about the spectrum of this transfer operator and thus about transversality. An important new feature of our method is that it is not global: the maps we consider are only required to be defined and holomorphic on a neighbourhood of some finite set. We will illustrate this method by obtaining transversality for a wide class of one-parameter families of interval and circle maps, for example for maps with flat critical points, but also for maps with complex analytic extensions such as certain polynomial-like maps. As in Tsujii's approach \cite{Tsu0,Tsu1}, for real maps we obtain {\em positive} transversality (where $>0$ holds instead of just $\ne 0$), and thus monotonicity of entropy for these families, and also (as an easy application) for the real quadratic family. This method additionally gives results for unimodal families of the form $x\mapsto |x|^\ell+c$ for $\ell>1$ not necessarily an even integer and $c$ real.
Given an integer $q\ge 2$ and a real number $c\in [0,1)$, consider the generalized Thue-Morse sequence $(t_n^{(q;c)})_{n\ge 0}$ defined by $t_n^{(q;c)} = e^{2πi c S_q(n)}$, where $S_q(n)$ is the sum of digits of the $q$-expansion of $n$. We prove that the $L^\infty$-norm of the trigonometric polynomials $σ_{N}^{(q;c)} (x) := \sum_{n=0}^{N-1} t_n^{(q;c)} e^{2πi n x}$, behaves like $N^{γ(q;c)}$, where $γ(q;c)$ is equal to the dynamical maximal value of $\log_q \left|\frac{\sin qπ(x+c)}{\sin π(x+c)}\right|$ relative to the dynamics $x \mapsto qx \mod 1$ and that the maximum value is attained by a $q$-Sturmian measure. Numerical values of $γ(q;c)$ can be computed.
We show that the graph of the classical Weierstrass function $\sum_{n=0}^\infty λ^n \cos (2πb^n x)$ has Hausdorff dimension $2+\logλ/\log b$, for every integer $b\ge 2$ and every $λ\in (1/b,1)$. Replacing $\cos(2πx)$ by a general non-constant $C^2$ periodic function, we obtain the same result under a further assumption that $λb$ is close to $1$.
In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps, maps with discontinuities but also for families of maps with complex analytic extensions such as certain polynomial-like maps.
We prove that any $C^{1+BV}$ degree $d \geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings.
For any polynomial map with a single critical point, we prove that its lower Lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle. Similar statement holds for the exponential maps and some other complex dynamical systems. We prove further that for the unicritical polynomials with positive area Julia sets almost every point of the Julia set has zero Lyapunov exponent. Part of this statement generalizes as follows: every point with positive upper Lyapunov exponent in the Julia set of an arbitrary polynomial is not a Lebegue density point.
For a real or complex one-dimensional map satisfying a weak hyperbolicity assumption, we study the existence and statistical properties of physical measures, with respect to geometric reference measures. We also study geometric properties of these measures.