arXiv subjects

Yuanyang Chang

Publications and source records attributed to Yuanyang Chang.

5 recordsLinked to original sources

arXiv2024

On orbit complexity of dynamical systems: intermediate value property and level set related to a Furstenberg problem

For symbolic dynamics with some mild conditions, we solve the lowering topological entropy problem for subsystems and determine the Hausdorff dimension of the level set with given complexity, where the complexity is represented by Hausdorff dimension of orbit closure. These results can be applied to some dynamical systems such as $\beta$-transformations, conformal expanding repeller, etc. We also determine the dimension of the Furstenberg level set, which is related to a problem of Furstenberg on the orbits under two multiplicatively independent maps.

math.DS↗

arXiv2024

Van der Corput and metric theorems for geometric progressions for self-similar measures

We prove a van der Corput lemma for non-atomic self-similar measures $\mu$. As an application, we show that the correlations of all finite orders of $( x^n \mod 1 )_{n\geq 1}$ converge to the Poissonian model for $\mu$-a.e. $x$, assuming $x>1$. We also complete a recent result of Algom, Rodriguez Hertz, and Wang (obtained simultaneously by Baker and Banaji), showing that any self-conformal measure with respect to a non-affine real analytic IFS has polynomial Fourier decay.

math.DS↗

arXiv2020

Lower Assouad type dimensions of uniformly perfect sets in doubling metric spaces

In this paper, we are concerned with the relationship among the lower Assouad type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as $θ$ tends to 1 equals to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as $θ$ tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi lower Assouad dimension.

math.CA↗

arXiv2018

Quantitative recurrence properties and homogeneous self-similar sets

Let $K$ be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map $T: K\rightarrow K$ induced by the shift. Let $μ$ be the natural self-similar measure supported on $K$. For a positive function $φ$ defined on $\mathbb{N}$, we show that the $μ$-measure of the following set \begin{equation*} R(φ):=\{x\in K: |T^n x-x|<φ(n) \; \text{for infinitely many} \; n\in\mathbb{N}\} \end{equation*} is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy law holds for the general Hausdorff measure, which completes the metric theory of this set.

math.DS↗

arXiv2017

Fourier decay bound and differential images of self-similar measures

In this note, we investigate $C^2$ differential images of the homogeneous self-similar measure associated with an IFS $\mathcal{I}=\{ρx+a_j\}_{j=1}^m$ satisfying the strong separation condition and a positive probability vector $\vec{p}$. It is shown that the Fourier transforms of such image measures have power decay for any contractive ratio $ρ\in (0, 1/m)$, any translation vector $\vec{a}=(a_1, \ldots, a_m)$ and probability vector $\vec{p}$, which extends a result of Kaufman on Bernoulli convolutions. Our proof relies on a key combinatorial lemma originated from Erdős, which is important in estimating the oscillatory integrals. An application to the existence of normal numbers in fractals is also given.

math.CA↗