◆ Fractal Geometry, Geometric Measure Theory
◆ Dynamic System and Ergodic Theory
◆ Metric Number Theory, Diophantine Approximation
◆ Random Covering
◆ Aug. 2011 - July 2013 Post-doctoral Fellow Department of mathematical Science, University of Oulu, Finland
◆ Aug. 2010 - present Faculty Department of mathematics, South China University of Technology, China
◆ Aug. 2009 - July 2010 Post-doctoral Fellow Department of mathematics, Taiwan University, Taiwan
◆ Oct. 30, 2016-Nov. 6, 2016 Visit Korea Institute for Advanced Study (KIAS), Seoul, South Korea
◆Sep. 1, 2016-Sep. 30, 2016 Visit National Center for Theoretical Sciences (Taipei), Taiwan
◆August 3, 2015-August 30, 2015 Visit Department of mathematical Science, University of Oulu, Finland
◆Sep. 1, 2014-Oct. 29, 2014 Visit Department of Mathematics, Chinese University of Hong Kong, Hong Kong
◆August 12, 2014-August 21, 2014 Visit Department of Mathematics Education, Dongguk University, South Korea
◆July 12, 2014-August 4, 2014 Visit National Center for Theoretical Sciences (Hsinchu), Taiwan
◆Feb. 24, 2014-March 9, 2014, Visit Department of Mathematics, Chinese University of Hong Kong, Hong Kong
◆May 30, 2013-June 5, 2013 Visit Department of Mathematics, University of Bremen, Germany
◆May 20, 2013-May 22, 2013 Visit Department of Mathematics, University of Jyväskylä, Finland
◆April 22, 2013-April 27, 2013 Visit Department of Mathematics, University of Bristol, UK
◆Oct. 1, 2012-Oct. 6, 2012 Visit Department of Mathematics, University of Helsinki, Finland
◆June 16, 2012-June 23, 2012 Visit Morningside Center of Mathematics, Chinese Academy of Science, China
◆Feb. 12, 2012-Feb. 18, 2012 Visit Department of Mathematics, University of Helsinki, Finland
◆June 3, 2011-July 2, 2011 Visit Department of Mathematics & CMTP, National Central University, Taiwan
◆Jan. 17, 2010 -Feb. 28, 2010 Visit Department of Statistics and Probability, Michigan State University, USA
◆ undergraduate courses: Mathematical Analysis (II), Real Analysis, Probability and Statistics (teaching in English) Calculus (I, II), Integral Transform
◆ graduate courses: Geometric Measure Theory, Topological Dynamical Systems, Ergodic Theory
(32) J. R. Tang, B. Li and W.-C. Cheng, Some properties on topological entropy of free semigroup action, Dyn. Syst., 33 (2018), no. 1, 54-71. [abstract] [link]
The aim of this paper is to examine the topological entropy for a free semigroup action defined by Bufetov using separated and spanning sets. First, this study reveals that such entropy is a topological conjugacy invariant and also can be equivalently defined usingopen covers. Furthermore, a quantitative analogue of Bowen’s theorem for semiconjugacy is provided and we compared the topological entropies presented by Bufetov and Bi´s. Finally, a formula for the entropy of skew-product transformation with respect to the subshiftis obtained.
(31) L.-X. Zheng, M. Wu and B. Li,The exceptional sets on the run-length function of beta-expansions, Fractals, 25(2017), no. 6, 1750060, 10pp. [abstract] [link]
Let \(\beta> 1\) and the run-length function \(r_n(x,\beta)\) be the maximal length of consecutive zeros amongst the first \(n\) digits in the \(\beta\)-expansion of \(x\in[0,1]\). The exceptional set \(E_{\max}^{\varphi}=\left\{x \in [0,1]:\liminf_{n\rightarrow \infty}\frac{r_n(x,\beta)}{\varphi(n)}=0, \limsup_{n\rightarrow \infty}\frac{r_n(x,\beta)}{\varphi(n)}=+\infty\right\}\) is investigated, where \(\varphi: \mathbb{N} \rightarrow \mathbb{R}^+\) is a monotonically increasing function with \(\lim\limits_{n\rightarrow \infty }\varphi(n)=+\infty\). We prove that the set \(E_{\max}^{\varphi}\) is either empty or of full Hausdorff dimension and residual in \([0,1]\) according to the increasing rate of \(\varphi\).
(30) A. Käenmäki and B. Li,Genericity of dimension drop on self-affine sets, Statist. Probab. Lett., 126(2017), 18-25. [abstract] [link]
We prove that generically, for a self-affine set in \(\mathbb{R}^d\), removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question.
(29) E. Järvenpää, M. Järvenpää, H. Koivusalo, B. Li, V. Suomala and Y. M. Xiao, Hitting probabilities of random covering sets in tori and metric spaces, Electronic J. Probab., 22(2017), no. 1, 1-18. [abstract] [link]
We provide sharp lower and upper bounds for the Hausdorff dimension of the intersectionof a typical random covering set with a fixed analytic set both in Ahlfors regularmetric spaces and in the d-dimensional torus. In metric spaces, we consider coveringsets generated by balls and, in tori, we deal with general analytic generating sets.
(28) W.-B. Liu and B. Li, Chaotic and topological properties of continued fractions, J. Number Theory., 174(2017), 369-383. [abstract] [link]
We prove that there exists a scrambled set for the Gauss map with full Hausdorff dimension, in particular, it is chaotic in the sense of Li–Yorke. Meanwhile, we also investigate the topological properties of the sets of points with dense or non-dense orbits.
(27) L.-X. Zheng, M. Wu and B. Li, The topological property of the irregular sets on the length of basic intervals in beta-expansions, J. Math. Anal. Appl.,499(2017),127-137. [abstract] [link]
Let \(\beta>1\) be a real number. A basic interval of order nis a set of real numbers in \((0, 1]\) having the same first ndigits in their \(\beta\)-expansion which contains \(x \in(0, 1]\), denote by \(I_{n}(x)\)and write the length of \(I_{n}(x)\) as \(|I_{n}(x)|\). In this paper, we prove that the extremely irregular set containing points \(x \in[0, 1]\) whose upper limit of \(\frac{−\log\beta}{|I_{n}(x)|}\) equals to \(1 +\lambda(\beta)\) is residual for every \(\lambda(\beta)>0\), where \(\lambda(\beta) \) is a constant depending on \(\beta\).
(26) D. H. Kim and B. Li, Zero-one law of Hausdorff dimensions of recurrent sets, Discrete Contin. Dyn. Syst.,36(2016), no. 10, 5477-5492.[abstract] [link]
Let \((\Sigma, \sigma)\) be the one-sided shift space with \(m\) symbols and \(R_n(x)\) be the first return time of \(x\in\Sigma\) to the \(n\)-th cylinder containing \(x\). Denote\(E^\varphi_{\alpha,\beta}=\left\{x\in\Sigma: \liminf_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}=\alpha,\ \limsup_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}=\beta\right\},\)where \(\varphi: \mathbb{N}\to \mathbb{R}^+\)is a monotonically increasing function and \(0\leq\alpha\leq\beta\leq +\infty\).We show that the Hausdorff dimension of the set \(E^\varphi_{\alpha,\beta}\) admits a dichotomy: it is either zero or one depending on \(\varphi, \alpha\) and \(\beta\).
(25) L.-L. Fang, M. Wu and B. Li，Limit theorems related to beta-expansion and continued fraction expansion, J. Number Theory.,163(2016), 385-405.[abstract] [link]
Let \( \beta >1\) be a real number and \( x \in [0, 1)\) be an irrational number. Denote by \(k_{n}(x)\)the exact number of partial quotients in the continued fraction expansion of \( x \) given by the first ndigits in the \(\beta \) -expansion of \(x(n \in\mathbb{N})\). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence \(\{k_n, n ≥1\}\), which generalize the results of Faivre [8]and Wu [31]respectively from \(\beta=10\) to any \(\beta>1\).
(24) A. Käenmäki, B. Li and V. Suomala, Local dimensions in Moran constructions, Nonlinearity.,29(2016), 807-822.[abstract] [link]
We study the dimensional properties of Moran sets and Moran measuresin doubling metric spaces. In particular, we consider local dimensions and \(L^q\)-dimensions. We generalise and extend several existing results in this area.
(23) E. Järvenpää, M. Järvenpää, B. Li and O. Stenflo, Random affine code tree fractals and Falcorner-Sloan condition, Ergodic Theory Dynam. Systems.,36(2016), 1516-1533.[abstract] [link]
We calculate the almost sure dimension for a general class of random affine codetree fractals in \(R^d\) . The result is based on a probabilistic version of the Falconer–Sloancondition \(C(s)\) introduced in Falconer and Sloan [Continuity of subadditive pressure forself-affine sets. Real Anal. Exchange 34 (2009), 413–427]. We verify that, in general,systems having a small number of maps do not satisfy condition \(C(s)\). However, thereexists a natural number \(n\) such that for typical systems the family of all iterates up to leve l\(n\) satisfies condition \(C(s)\).
(22) J.-J. Li and B. Li, Hausdorff dimension of some irregular sets associated with beta-expansions, Sci. China Math., 59(2016), no. 3, 445-458.[abstract] [link]
The Hausdorff dimensions of some refined irregular sets associated with \(\beta\)-expansions are determined for any \(\beta>1\). More precisely, Hausdorff dimensions of the sets\(\left\{x\in[0,1):\liminf\limits_{n\rightarrow\infty}\frac{S_{n}(x,\beta)}{n}=\alpha_{1},\limsup\limits_{n\rightarrow\infty}\frac{S_{n}(x,\beta)}{n}=\alpha_{2}\right\}, \alpha_1,\alpha_2\ge 0\)are obtained completely, where \(S_{n}(x,\beta)=\sum_{k=1}^{n}\varepsilon_k(x,\beta) \) denotes the sum of the first \(n\) digits of the \(\beta\)-expansion of \(x\). As an application, we present another concise proof of that the set of points \(x\in [0,1)\) satisfying \(\lim\limits_{n\to\infty}\frac{S_n(x,\beta)}{n}\) does not exist is of full Hausdorff dimension.
(21) B. Li, T. Sahlsten and T. Samuel, Intermediate beta-shifts of finite type, Discrete Contin. Dyn. Syst., 36(2016), no. 1, 323-344.[abstract] [link]
An aim of this article is to highlight dynamical dierences betweenthe greedy, and hence the lazy, \(\beta\)-shift (transformation) and an intermediate \(\beta\)-shift (transformation), for a fixed \( \beta \in (1; 2)\). Specically, a classification in terms of the kneading invariants of the linear maps \(T_{\beta, \alpha} : x\mapsto \beta x+\alpha\ \text{mod}\ 1\)for which the corresponding intermediate \(\beta\)-shift is of fnite type is given. This characterisation is then employed to construct a class of pairs \((\beta, \alpha)\) such that the intermediate \(\beta\)-shift associated with \(T_{\beta,\alpha}\) is a subshift of fnite type. It is also proved that these maps \(T_{\beta, \alpha}\) are not transitive. This is in contrast to the situation for the corresponding greedy and lazy \(\beta\)-shifts and \(\beta\)-transformations,for which both of the two properties do not hold.
(20) L.-L. Fang, M. Wu, N.-R. Shieh and B. Li, Random Continued fractions: Levy constant and Chernoff-type estimate, J. Math. Anal. Appl., 429(2015), no. 1, 513-531.[abstract] [link]
Given a stochastic process \(\{A_{n}, n ≥1\} \) taking values in natural numbers, the random continued fractions aredefined as \([A_1, A_2, ···, A_n, ···]\) analogously to the continued fraction expansion of real numbers. Assume that\(\{A_n, n≥1\}\) is ergodic and the expectation \(E(logA_{1})<\infty\), we give a Lévy-type metric theorem which covers that of real case presented by Lévy in 1929. Moreover, a corresponding Chernoff-type estimate is obtained under the conditions that\(\{A_n, n≥1\}\) is \(\psi\)-mixing and for each \( 0 <t <1\), \(E(A_t^1) <\infty\).
(19) M.-M. Kong, W.-C. Cheng and B. Li, Topological pressure for non autonomous systems, 76(2015), 82-92, Chaos Solitons Fractals,, 76(2015), 82-92,[abstract] [link]
Kolyada and Snoha (1996) proposed the notion of entropy-like invariants for nonautonomousdynamical systems. This paper was based on that concept and involved extendingthe behavior of topological pressure to a fixed sequence of maps. Specifically, this studyinvestigated how the pressure changes when the potentials or the mappings vary. The analoguesof basic properties were obtained, and this study also reveals that, for any continuousmaps \(T\) and \(S\) from a compact metric space into itself, the maps \(T \circ S\) and\(S \circ T\) have thesame topological pressure (with respect to the corresponding potential functions).
(18) C.-H. Chen, H. Koivusalo, B. Li and V. Suomala, Projections of random covering sets, J. Fractal Geom. J. Fractal Geom., 1 (2014), no. 4, 449–467,[abstract] [link]
We show that, almost surely, the Hausdorff dimension \(s_0 \) of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension \(k>s_0\). The result holds for random covering sets with a generating sequence of ball-like sets, and is obtained by investigating orthogonal projections of a class of random Cantor sets.
(17) J.-C. Ban and B. Li, The multifractal spectra for the recurrence rates of beta-transformations, J. Math. Anal. Appl., 420(2014), No. 2, 1662-1679.[abstract] [link]
In this paper, we show a handy approximate approach to provide a lower bound of the Hausdorff dimension of a given subset in\([0, 1) \) related to β-transformation dynamical system. Here approximation means from special class with \(\beta\) shift satisfying the specification property or being subshift of finite type to general \(\beta >1\). As an application, we obtain the multifractal spectra for the recurrence rate of the first return time of \(\beta\) -transformation, including the cases returning to the ball and cylinder.
(16) R. Kuang, W.-C. Cheng, D.-K. Ma and B. Li, Different forms of entropy dimension for zero entropy systems, Dyn. Syst., 29(2014), No.2, 239-254.[abstract] [link]
The aim of this paper is to introduce the lower \(s\)-topological entropy to distinguish zero entropy systems. This quantity is an invariant factor under topological conjugacy and power rule can be shown. Some examples are given to show that the lower entropy dimension can attain any value in \((0,1)\), and are different with the upper one and the entropy dimension in the sense of Bowen. A counterexample is used to indicate that the product rule does not hold, and the lower \(s\)-topological entropy of the subsystem for the non-wandering set can be strictly less than that of the system when \(0<s<1\). Finally, this study also constructs a dynamical system to show that the transitive system with zero entropy dimension may not be minimal.
(15) B. Li and V. Suomala, A note on the hitting probabilities of random covering sets, Ann. Acad. Sci. Fenn. Math., 39(2014), 625-633.[abstract] [link]
Let \(E=\limsup\limits_{n\to\infty}(g_n+\xi_n)\) be the random covering set on the torus \(\mathbb{T}^d\), where \(\{g_n\}\) is a sequence of ball-like sets and \(\xi_n\) is a sequence of i.i.d. random variables uniformly distributed on \(T^d \). We prove that \(E\cap F\neq\emptyset\) almost surely whenever the Hausdorff dimension of the analytic set \(F\), \(\dim_H(F)>d-\alpha\), where \(\alpha\) is the almost sure Hausdorff dimension of \(E\). Moreover, examples are given to show that the condition on \(\dim_H(F) \) can not be replaced by the packing dimension of \(F\).
(14) B. Li, T. Persson, B. Wang and J. Wu, Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions, Math. Z, 176(2014), 799-827.[abstract] [link]
We consider the distribution of the orbits of the number 1 under the \(\beta\)-transformations \(T_\beta\) as \(\beta\) varies. Mainly, the size of the set of \(\beta > 1\) for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension.The dimension of the following set \( E(\{\ell_n\}_{n\geq 1},x_0):=\{\beta>1:|T^n_\beta (1)-x_0|<\beta^{-\ell_n} \text{for infinitely many}\ n\in\Bbb N\} \) is determined, where \(x_0\) is a given point in \([0, 1]\) and \({\ell_n}n≥1\) is a sequence of integerstending to infinity as \(n →∞\). For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of \(\beta\) with a common prefix in the expansion of 1) in the parameter space \(\{\beta \in \mathbb{R} : \beta > 1 \}\).
(13) B. Li, B.-W. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. London Math. Soc. (3), 108(2014),159-186.[abstract] [link]
Let \(([0, 1), T)\) be the dynamical system of continued fractions. Let\(\{z_n\}_{n \geq 1}\) be a sequenceof real numbers in \([0, 1]\) and \(\psi: \mathbb{N} \times [0, 1) → \mathbb{R}_+\) be a positive function. A point \(x \in [0, 1)\) is said to be \(\psi\)-approximable by \(\{z_n\}_{n \geq1}\) if \(|T_{n} x − z_n| \) < ψ(n, x) holds for infinitely many \(n \in \mathbb{N}\). In this paper, the Hausdorff dimension of the set of \(\psi\)-approximable points is studied. The dimensions are completely determined when \(\psi(n, x)=\psi(n)\) independent on \(x\) and when \(\psi(n, x) = e^{-(f(x)+···+f(T^{n−1}x)} \) with \( f \) a positive continuous function. For the proof of these results, a relationship between a ball in\([0, 1)\) and the cylinders defined by the partial quotientsin continued fractions is investigated. It is shown that a ball can be sufficiently packed by cylinders of the same order and of comparable length, which gives us explicit continued fraction representations in locating the points in a ball in \([0, 1)\).
(12) B. Li, N.-R. Shieh and Y.-M. Xiao, Hitting probabilities of the random covering sets, Fractal geometry and dynamical systems in pure and applied mathematics. II. Fractals in applied mathematics, 307-323, Contemp. Math. 601, Amer. Math. Soc., Providence, RI, 2013. Contemp. Math. 601, Amer. Math. Soc., Providence, RI. (3), 2013.[abstract] [link]
Let \(E\) be the Dvoretzky random covering sets on the circle. By applyingthe method of limsup type random fractals, asillustrated in Khoshnevisan, Peres and Xiao , wedetermine the hitting probability \(P(E\cap G\ne \emptyset)\) and thepacking dimension of the intersection \(E \cap G\), where \(G\) is anarbitrary Borel set on the circle.
(11) X.-H. Hu, B. Li and J. Xu, Metric theorem and Hausdorff dimension on recurrence rate of Laurent series, Bull. Korean Math.(3), 51(2014), No.1, 157-171[abstract] [link]
We show that the recurrence rates of Laurent series aboutcontinued fractions almost surely coincide with their pointwise dimensionsof the Haar measure. Moreover, let \(E_{\alpha,\beta} \) be the set of points with lowerand upper recurrence rates \(\alpha,\beta(0 \leq \alpha\leq \beta \leq \infty)\), we prove that all thesets\(E_{\alpha,\beta} \) are of full Hausdorff dimension. Then the recurrence sets \(E_{\alpha,\beta} \) have constant multifractal spectra.
(10) E. Järvenpää, M. Järvenpää, H. Koivusalo, B. Li and V. Hausdorff dimension of affine random covering sets in torus, Ann. Ann. Inst. Henri Poincare Probab. Stat(3), 50(2014) no.4, 1371-1384.[abstract] [link]
We calculate the almost sure Hausdorff dimension of the random covering set \( lim sup_{n→∞}(g_n + ξ_n) \)in \(d\)-dimensional torus \( T^d\), where the sets \(g_n \subset T^d \) are parallelepipeds, or more generally, linear images of a set with nonempty interior, and \(ξ_n\in T_d\)are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linearmappings, holds provided that the sequences of the singular values are decreasing.
(9) W.-C. Cheng and B. Li,Topological pressure dimension, Chaos Solitons Fractals, 53(2013), 10-17.[abstract] [link]
This paper presents the properties of topological pressure dimension, which is an extensionof the entropy dimension. Specifically, this paper studies the relationships among differenttypes of topological pressure dimension and identifies an inequality relating them. Thisanalysis calculates analogs of many known results of topological pressure. In particular,we will show the value of the pressure dimension is always equal to or greater than 1for any positive constant potential function.
(8) R. Kuang, W.-C. Cheng and B. Li,Fractal entropy of nonautonomous systems, Pacific J. Math, 262(2013), no. 2, 421-436.[abstract] [link]
We define formulas of entropy dimension for a nonautonomous dynamicalsystem consisting of a sequence of continuous self-maps of a compact metricspace. This study reveals analogues of basic propositions for entropydimension, such as the power rule, product rule and commutativity, etc.These properties allow us to convert to an equality an inequality found byde Carvalho (1997) concerning the product rule for the autonomous dynamicalsystem. We also prove a subadditivity rule of entropy dimension forone-dimensional dynamics based on our previous work.
(7) D.-K. Ma, R. Kuang and B. Li, Topological entropy dimension for noncompact sets, Dyn. Syst, 262(2013), 27(2012), No. 3, 303-316.[abstract] [link]
Similar with the fractal dimension, we introduce the concept of topologicalentropy dimension to classify the sets with entropy zero. We prove that theentropy dimension of the space in this article is not greater than that defined byDe Carvalho, where he introduced the entropy dimension for the system, and givesome examples indicating that such inequality is optimal. Some basic propositionsof entropy dimension are discussed and it turns out that the entropy dimension isinvariant under conjugacy. The property of the countable stability and a powerrule for the entropy dimension of any set are obtained. It is shown that any setshares the same entropy dimension with its image set.
(6) B. Li and Y.-C. Chen, Chaotic and topological properties of \(beta\)-transformations, J. Math. Anal. Appl. , 383(2011), 585-596.[abstract] [link]
In this paper, we prove that the \(\beta\)-transformations are chaotic in the sense of both Li–Yorke and Devaney. The topological and metric properties of the sets of points with denseor non-dense orbits are investigated. We also prove the result that the set of points withnon-dense orbits under the \(\beta\)-transformation is of full Hausdorff dimension for any \(\beta>1.\)
(5) A. H. Fan, T. Langlet and B. Li, Quantitative uniform hitting in exponentially mixing systems, Recent developments in fractals and related fields, Appl. Numer. Harmon. Anal, Birkhäuser Boston, Inc,, Boston, MA, 2010[abstract] [link]
Consider an exponentially mixing metric measure preserving system \((X,B, \mu, T,d)\). Let \(\alpha_{\max}\) be the maximal local dimension of \(\mu\). It is proved that if \(\tau<1/\alpha_{\max}\), then for \(\mu\)-almost all \(x\) and for every \(y\in X\) we have \(\liminf_{n\to \infty} n^\tau d(T^n x, y)=0\). The critical value \(1/\alpha_{\max}\) is optimal in manycase.
(4) W.-C. Cheng and B. Li,Zero entropy systems, J. Stat. Phys. , 140(2010), no. 5, 1006–1021.[abstract] [link]
This paper introduces the notion of entropy dimension to measure the complexityof zero entropy dynamical systems, including the probabilistic and the topological versions.These notions are isomorphism invariants for measure-preserving transformation and continuity.We discuss basic propositions for entropy dimension and construct some examples toshow that the topological entropy dimension attains any value between 0 and 1. This paperalso gives a symbolic subspace to achieve zero topological entropy, but with full entropydimension.
(3) B. Li and J. Wu, Beta-expansion and continued fraction expansion over formal Laurent series, Finite Fields Appl , 14(2008), no. 3, 635–647. [abstract] [link]
Let \(x\in I \) be an irrational element and \(n\geq 1\), where \(I\) is the unit disc in the field of formal Laurent series \(F((X^{-1}))\), we denote by \(k_n(x)\) the number of exact partial quotients incontinued fraction expansion of \(x\), given by the first \(n\) digitsin the \(\beta\)-expansion of \(x\), both expansions are based on\(F((X^{-1}))\). We obtain that\(\liminf_{n\to+\infty}{\frac{k_n(x)}{n}}=\frac{\deg \beta}{2Q^\ast(x)}, \limsup_{n\to +\infty}{\frac{k_n(x)}{n}}=\frac{\deg\beta}{2Q_\ast(x)},\) where \(Q^\ast(x), Q_\ast(x)\) are the upper andlower constants of \(x\) respectively. Also, a central limit theoremand an iterated logarithm law for \(\{k_n(x)\}_{n\geq 1}\) areestablished.
(2) B. Li, J. Wu and J. Xu,Metric properties and exceptional sets of beta-expansions over formal series, Monatsh. Math, 155(2008), no. 2, 145–160. [abstract] [link]
This paper is concerned with the metric properties of \(\beta\)-expansions over the field of formal Laurent series. We will see that there are essential differences between \(\beta\)-expansions of the formal Laurent series case and the classical real case. Also the Hausdorff dimensions of some exceptional sets, with respect to the Haar measure, are determined.
(1) B. Li and J. Wu, Beta-expansion and continued fraction expansion, J. Math. Anal. Appl, 339(2008), no. 2, 1322–1331[abstract] [link]
For any real number \(\beta>1\), let \(\varepsilon(1, \beta)=\big(\varepsilon_{1}(1),\varepsilon_{2}(1), \cdots, \varepsilon_{n}(1), \cdots\big)\) be theinfinite \(\beta \)-expansion of 1. Define \(l_{n}=\sup\{k\geq0:\varepsilon_{n+j}(1)=0,\) for all \( 1\leq j\leq k\}\). Let\(x\in[0,1)\) be an irrational number. We denote by \(k_n(x)\) theexact number of partial quotients in continued fraction expansion of\(x\) given by the first \(n\) digits in the \(\beta\)-expansion of \(x\).If \(\limsup\limits_{n\to +\infty}{l_n}\) is finite, we obtain thatfor all \({x\in [0, 1)\backslash \mathbb{Q}}\),\(\liminf_{n\to+\infty}{\frac{k_n(x)}{n}}=\frac{\log\beta}{2\beta^\ast(x)}, \limsup_{n\to+\infty}{\frac{k_n(x)}{n}}=\frac{\log\beta}{2\beta_\ast(x)},\) where\(\beta^\ast(x)\), \(\beta_\ast(x)\) are the upper and lower Levy constants, which generalize the result in[ J. Wu, Continued fraction and decimalexpansions of an irrational number, Adv. Math. 206 (2) (2006) 684–694 ].Moreover, if \(\limsup\limits_{n\to+\infty}{\frac{l_n}{n}}=0\), we also get the similar result except a small set.
(46) Nov. 1, 2016, Chaotic and topological properties of continued fractions, Korea Institute for Advanced Study (KIAS), Korea
(45) Oct. 21-25, 2016, Thermodynamic formalism for the singular value function,2016 National Conference on Dynamical Systems, Guangxi University of Finance and Economics, Nanning, China
(44) June 20-29, 2016, Zero-one law of Hausdorff dimension of the recurrent sets，2016 Summer School on Fractal Geometry and Complex Dimensions, California Polytechinic State University, San Luis Obispo, USA
(43) Sep. 19-25, 2015, Hitting probabilities of random covering sets in higher dimension,Fractals and related fields III, Porquerolles, France
(42) Sep. 18, 2015, Random affine code tree fractals and Falconer-Sloan condition, Institute Henri Poincaré, Paris, France
(41) August 24, 2015, Some topics on random covering sets over the torus, University of Oulu, Oulu, Finland
(40) June 10-12, 2015 Diophantine approximation in parameter spaces of the dynamical system of beta-transformations，Fractals and Numeration, Admont，Austria
(39) March 14-15, 2015 Hitting probabilities of random covering sets in high dimension，2015 AMS Central Section Meeting－Special Section on Fractal and Tilings，Michigan State University，USA
(38) Oct. 17, 2014, Dvoretzkoy random covering and potential theory, Chinese University of Hong Kong, Hong Kong
(37) August 8-12, 2014 Intermediate beta-shifts of finite type, ICM2014 Satellite conference on dynamical systems and related topics, Chungnam National University, Daejeon, South Korea.
(36) July 16, 2014, Diophantine approximation in parameter spaces of the dynamical system of beta-transformations, Institute of Mathematics, Academia Sinica, Taipei, Taiwan
(35) March. 7, 2014, The shrinking target problem and continued fractions, Chinese University of Hong Kong, Hong Kong
(34) Dec. 20-21, 2013 Random covering sets in high dimension, Workshop on fractal geometry and related topics, Sun Yat-Sen University, Guangzhou, China
(33) Dec. 16, 2013 Some topics on the Hausdorff dimensions of the random covering sets, School of Mathematics and Statistics, Huazhong University of Science of Technology, Wuhan, China
(32) June 11-14, 2013 The shrinking problem in the dynamical system of continued fractions, Continued fractions, Interval exchanges and Applications to Geometry, Pisa, italy
(31) June 4, 2013 The dimension theory of well-approximable sets and continued fractions, colloquium, University of Bremen, Germany
(30) April 25, 2013 The shrinking target problem in the dynamical system of continued fractions, Seminars, University of Bristol, UK
(29) March 18-22, 2013 Diophantine approximation of the orbit of 1 in the dynamical system of beta-expansions,Bremen Winter School on Multifractals and Number Theory, University of Bremen, Gemany
(28) Dec. 10-14, 2012 Diophantine approximation of the orbit of 1 in beta-transformation dynamical system, International conference on advance on fractals and related topics, Chinese University of Hong Kong, Hong Kong
(27) Oct. 4, 2012 The distribution of the orbits in Beta-transformation dynamical system, Department of Mathematics, University of Helsinki, Helsinki, Finland
(26) April 23-27, 2012 Hitting probabilities of the random covering sets, Ergodic methods in dynamics, Bedlewo, Poland
(25) April 16-20, 2012 The multifractal spectra for the recurrence rates of beta-transformations, Ergodic Theory and Dynamical Systems: Perspectives and Prospects, University of Warwick, UK
(24) Feb. 16, 2012 Random covering problems on the circle and for dynamical systems, Department of Mathematics, University of Helsinki, Helsinki, Finland
(23) Sep. 12, 2011 Some topics on random covering problem, Colloquium, Department of Mathematical Science, University of Oulu, Oulu, Finland
(22) June 24, 2011 Metric and topological properties of beta-expansionas, Department of Mathematics, National Central University, Jongli City, Taiwan
(21) June 13, 2011 Some topics on entropy dimension, Dynamical system seminar, Institute of Mathematics, Academia Sinica, Taipei, Taiwan
(20) May 27-31, 2011 Some topics on beta-expansions, 2011 National Mathematical Conference on Fractal Theory and Dynamic Systems, Zhangjiajie, China
(19) Dec. 12-16, 2010 Quantitative recurrence in exponentially mixing systems.
International Conference on Nonlinear Analysis, South China University of Technology, Guangzhou, China
(18) Nov. 26, 2010 Some topics on entropy dimensions, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China
(17) August 20-23, 2010 Random covering and recurrence problems. Summer School on complex analysis and fractal geometry, Hunan Normal University, Changsha, China
(16) June 9, 2010 Some dynamics related to the number theory and applications of Birkhoff ergodic theorem. Colloquium, Tunghai University, Taichung, Taiwan
(15) May 12-May 15, 2010 Recurrence and hitting problems of continued fraction and beta-shift dynamical systems. NCTS 2010 Workshop on Dynamical Systems, National Tsing-Hua University, Hsinchu, Taiwan
(14) May 4, 2010 Quantitative recurrence and uniform hitting of dynamical systems. National Dong Hwa University, Hualien, Taiwan
(13) April 23, 2010 An introduction to \(\beta\)--expansion and symbolic dynamics III. Dynamical system seminar, Institute of Mathematics, Academia Sinica, Taipei, Taiwan
(12) April 9, 2010 An introduction to \(\beta\)--expansion and symbolic dynamics II. Dynamical system seminar,Institute of Mathematics, Academia Sinica, Taipei, Taiwan
(11) March 26, 2010 An introduction to \(\beta\)--expansion and symbolic dynamics I. Dynamical system seminar, Institute of Mathematics, Academia Sinica, Taipei, Taiwan
(10) March 24, 2010 Recurrence and hitting rate of a class of dynamical systems. Mini-Workshop on Applied Analysis and Probability, Taiwan University, Taipei, Taiwan
(9) March 19, 2010 Recurrence and hitting problem in dynamical system. NCTS & CMMSC (NCTU) Seminar on Dynamical Systems, National Chiao Tung University, Hsinchu, Taiwan
(8) Dec. 4-Dec. 6 2009 Quantitative recurrence problem in dynamical systems. Annal Meeting of the Mathematical Society, National Chung Cheng University, Chiayi, Taiwan
(7) Oct. 21-Oct. 22 2009 Dynamical Systems Viewpoint of Continued Fraction and $\beta$-expansions. Workshop on Theory and Applications of Mathematical Analysis, Taiwan University, Taipei, Taiwan
(6) September 25, 2009 Probability in Symbolic Dynamics. NCTS/TPE & TIMS Joint Activity in Probability, Taiwan University, Taipei, Taiwan
(5) September 23, 2009 Some classical dynamical systems and applications of Birkhoff ergodic theorem. Colloquium, Fu Jen University, Taipei, Taiwan
(4) March 27, 2009 A relationship between \(\beta\)-shift and continued fraction dynamical system. Xiamen University, Xiamen, China
(3) March 28, 2008 R\'{e}currence uniforme pour les syst\`{e}mes dynamiques. Seminar of Doctors, LAMFA, Amiens, France
(2) October 11, 2007 Relationship between the beta-digits and the partial quotient of real numbers. Seminar on Probability and Ergodic Theory, Amiens, France
(1) June 23-June 25, 2007 Beta-expansions and continued fraction expansion. National Conference on Fractal and Dynamical Systems, Jiangsu University, Zhenjiang, China
School of Mathematics
South China University of Technology
Guangzhou, 510641
P. R. China
Office: 4314, Building #4
Email: scbingli@scut.edu.cn
李兵，男，华南理工大学教授、博士生导师。2013年入选“千百十工程“第七批校级培养对象,2016年入选“广东特支计划” 百千万工程青年拔尖人才。
◆ 2005年9月—2009年7月 武汉大学数学与统计学院和法国亚眠大学LAMFA实验室（CNRS UMR 6140）联合培养博士生，获得两校博士学位
◆ 2003年9月—2005年7月 武汉大学数学与统计学院读硕士（硕博连读）
◆ 1999年9月—2003年7月 武汉大学数学基地班读大学，获学士学位
◆ 2010年8月—现在 华南理工大学数学系任教
◆ 2015年 被遴选为博士生导师
◆ 2014年9月 破格晋升为教授
◆ 2011年 被遴选为硕士生导师
◆ 2010年12月 通过华南理工大学引进人才专业技术职务评审小组评审，被聘为副教授
◆ 2011年8月—2013年7月 芬兰Oulu大学数学科学系博士后
◆ 2009年8月—2010年7月 台湾大学数学系博士后
◆ 主讲本科生课程：数学分析（二）、实变函数、概率论与数理统计（全英）、高等数学（上、下）、积分变换
◆ 主讲研究生课程：拓扑动力系统、遍历理论、几何测度论
◆ 获得华南理工大学2014-2015学年度本科教学优秀二等奖
◆ 被评为数学学院2015年度优秀班主任
◆ 10、自仿集和随机自仿覆盖集的维数及相关问题研究(11671151)，国家自然科学基金面上项目，2017年1月2020年12月，主持，在研。
◆ 9、动力系统联立轨道渐近行为相关问题的研究(2018B0303110005)，广东省自然科学基金重点项目，2018年5月至2021年4月，主持，在研。
◆ 8、动力系统中的分形问题(2015ZZ055)，中央高校基本科研业务费 (重点项目)，2015年1月至2016年12月，主持，已结题。
◆ 7、动力系统中的熵和维数问题(2014A030313230)，广东省自然科学基金自由申请项目，2015年1月至2018年1月，主持，在研。
◆ 6、加倍度量空间中的重分形分析和次可加动力系统的广义不可约性(11411130372)，国家自然科学基金国际合作交流项目，2014年11月至2015年2月，主持，已结题。
◆ 5、分形及动力系统中的测度和维数理论(2013ZZ0085)，中央高校基本科研业务费 (重点项目)，2013年1月至2014年12月，主持，已结题。
◆ 4、高维随机覆盖问题及其在动力系统中的应用(11201155)，国家自然科学基金青年基金 2013 年1月至2015年12月，主持，已结题。
◆ 3、随机覆盖模型中的分形问题及相关分析，教育部留学回国人员启动基金 2013 年3月至2014年3月，主持，已结题。
◆ 2、与实数非整数基表示相关的若干分形问题(11126071)，国家自然科学基金数学天元基金项目，2012 年1月至2012 年12月，主持，已结题。
◆ 1、随机分形中的维数研究(2011ZM0083)， 中央高校基本科研业务费 (面上项目)，2011年1月至2012年12月，主持，已结题。
(32) J. R. Tang, B. Li and W.-C. Cheng, Some properties on topological entropy of free semigroup action, Dyn. Syst., 33 (2018), no. 1, 54-71. [摘要] [链接]
The aim of this paper is to examine the topological entropy for a free semigroup action defined by Bufetov using separated and spanning sets. First, this study reveals that such entropy is a topological conjugacy invariant and also can be equivalently defined usingopen covers. Furthermore, a quantitative analogue of Bowen’s theorem for semiconjugacy is provided and we compared the topological entropies presented by Bufetov and Bi´s. Finally, a formula for the entropy of skew-product transformation with respect to the subshiftis obtained.
(31) L.-X. Zheng, M. Wu and B. Li, The exceptional sets on the run-length function of beta-expansions, Fractals, 25(2017), no. 6, 1750060, 10pp. [摘要] [链接]
Let \(\beta> 1\) and the run-length function \(r_n(x,\beta)\) be the maximal length of consecutive zeros amongst the first \(n\) digits in the \(\beta\)-expansion of \(x\in[0,1]\). The exceptional set \(E_{\max}^{\varphi}=\left\{x \in [0,1]:\liminf_{n\rightarrow \infty}\frac{r_n(x,\beta)}{\varphi(n)}=0, \limsup_{n\rightarrow \infty}\frac{r_n(x,\beta)}{\varphi(n)}=+\infty\right\}\) is investigated, where \(\varphi: \mathbb{N} \rightarrow \mathbb{R}^+\) is a monotonically increasing function with \(\lim\limits_{n\rightarrow \infty }\varphi(n)=+\infty\). We prove that the set \(E_{\max}^{\varphi}\) is either empty or of full Hausdorff dimension and residual in \([0,1]\) according to the increasing rate of \(\varphi\).
For any real number \(\beta>1\), let \(\varepsilon(1, \beta)=\big(\varepsilon_{1}(1),\varepsilon_{2}(1), \cdots, \varepsilon_{n}(1), \cdots\big)\) be theinfinite \(\beta \)-expansion of 1. Define \(l_{n}=\sup\{k\geq0:\varepsilon_{n+j}(1)=0,\) for all \( 1\leq j\leq k\}\). Let\(x\in[0,1)\) be an irrational number. We denote by \(k_n(x)\) theexact number of partial quotients in continued fraction expansion of\(x\) given by the first \(n\) digits in the \(\beta\)-expansion of \(x\).If \(\limsup\limits_{n\to +\infty}{l_n}\) is finite, we obtain thatfor all \({x\in [0, 1)\backslash \mathbb{Q}}\),\(\liminf_{n\to+\infty}{\frac{k_n(x)}{n}}=\frac{\log\beta}{2\beta^\ast(x)},\ \ \\limsup_{n\to+\infty}{\frac{k_n(x)}{n}}=\frac{\log\beta}{2\beta_\ast(x)},\) where\(\beta^\ast(x)\), \(\beta_\ast(x)\) are the upper and lower Levy constants, which generalize the result in[ J. Wu, Continued fraction and decimalexpansions of an irrational number, Adv. Math. 206 (2) (2006) 684–694 ].Moreover, if \(\limsup\limits_{n\to+\infty}{\frac{l_n}{n}}=0\), we also get the similar result except a small set.
(30) A. Käenmäki and B. Li, Genericity of dimension drop on self-affine sets, Statist. Probab. Lett., 126(2017), 18-25. [摘要] [链接]
We prove that generically, for a self-affine set in \(\mathbb{R}^d\), removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question.
(29) E. Järvenpää, M. Järvenpää, H. Koivusalo, B. Li, V. Suomala and Y. M. Xiao, Hitting probabilities of random covering sets in tori and metric spaces, Electronic J. Probab., 22(2017), no. 1, 1-18. [摘要] [链接]
We provide sharp lower and upper bounds for the Hausdorff dimension of the intersectionof a typical random covering set with a fixed analytic set both in Ahlfors regularmetric spaces and in the d-dimensional torus. In metric spaces, we consider coveringsets generated by balls and, in tori, we deal with general analytic generating sets.
(28) W.-B. Liu and B. Li, Chaotic and topological properties of continued fractions, J. Number Theory., 174(2017), 369-383. [摘要] [链接]
We prove that there exists a scrambled set for the Gauss map with full Hausdorff dimension, in particular, it is chaotic in the sense of Li–Yorke. Meanwhile, we also investigate the topological properties of the sets of points with dense or non-dense orbits.
(27) L.-X. Zheng, M. Wu and B. Li, The topological property of the irregular sets on the length of basic intervals in beta-expansions, J. Math. Anal. Appl.,499(2017),127-137. [摘要] [链接]
Let \(\beta>1\) be a real number. A basic interval of order nis a set of real numbers in \((0, 1]\) having the same first ndigits in their \(\beta\)-expansion which contains \(x \in(0, 1]\), denote by \(I_{n}(x)\)and write the length of \(I_{n}(x)\) as \(|I_{n}(x)|\). In this paper, we prove that the extremely irregular set containing points \(x \in[0, 1]\) whose upper limit of \(\frac{−\log\beta}{|I_{n}(x)|}\) equals to \(1 +\lambda(\beta)\) is residual for every \(\lambda(\beta)>0\), where \(\lambda(\beta) \) is a constant depending on \(\beta\).
(26) D. H. Kim and B. Li, Zero-one law of Hausdorff dimensions of recurrent sets, Discrete Contin. Dyn. Syst.,36(2016), no. 10, 5477-5492.[摘要] [链接]
Let \((\Sigma, \sigma)\) be the one-sided shift space with \(m\) symbols and \(R_n(x)\) be the first return time of \(x\in\Sigma\) to the \(n\)-th cylinder containing \(x\). Denote\(E^\varphi_{\alpha,\beta}=\left\{x\in\Sigma: \liminf_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}=\alpha,\ \limsup_{n\to\infty}\frac{\log R_n(x)}{\varphi(n)}=\beta\right\},\)where \(\varphi: \mathbb{N}\to \mathbb{R}^+\)is a monotonically increasing function and \(0\leq\alpha\leq\beta\leq +\infty\).We show that the Hausdorff dimension of the set \(E^\varphi_{\alpha,\beta}\) admits a dichotomy: it is either zero or one depending on \(\varphi, \alpha\) and \(\beta\).
(25) L.-L. Fang, M. Wu and B. Li，Limit theorems related to beta-expansion and continued fraction expansion, J. Number Theory.,163(2016), 385-405.[摘要] [链接]
Let \( \beta >1\) be a real number and \( x \in [0, 1)\) be an irrational number. Denote by \(k_{n}(x)\)the exact number of partial quotients in the continued fraction expansion of \( x \) given by the first ndigits in the \(\beta \) -expansion of \(x(n \in\mathbb{N})\). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence \(\{k_n, n ≥1\}\), which generalize the results of Faivre [8]and Wu [31]respectively from \(\beta=10\) to any \(\beta>1\).
(24) A. Käenmäki, B. Li and V. Suomala, Local dimensions in Moran constructions, Nonlinearity.,29(2016), 807-822.[摘要] [链接]
We study the dimensional properties of Moran sets and Moran measuresin doubling metric spaces. In particular, we consider local dimensions and \(L^q\)-dimensions. We generalise and extend several existing results in this area.
(23) E. Järvenpää, M. Järvenpää, B. Li and O. Stenflo, Random affine code tree fractals and Falcorner-Sloan condition, Ergodic Theory Dynam. Systems.,36(2016), 1516-1533.[摘要] [链接]
We calculate the almost sure dimension for a general class of random affine codetree fractals in \(R^d\) . The result is based on a probabilistic version of the Falconer–Sloancondition \(C(s)\) introduced in Falconer and Sloan [Continuity of subadditive pressure forself-affine sets. Real Anal. Exchange 34 (2009), 413–427]. We verify that, in general,systems having a small number of maps do not satisfy condition \(C(s)\). However, thereexists a natural number \(n\) such that for typical systems the family of all iterates up to leve l\(n\) satisfies condition \(C(s)\).
(22) J.-J. Li and B. Li, Hausdorff dimension of some irregular sets associated with beta-expansions, Sci. China Math., 59(2016), no. 3, 445-458.[摘要] [链接]
The Hausdorff dimensions of some refined irregular sets associated with \(\beta\)-expansions are determined for any \(\beta>1\). More precisely, Hausdorff dimensions of the sets\(\left\{x\in[0,1):\liminf\limits_{n\rightarrow\infty}\frac{S_{n}(x,\beta)}{n}=\alpha_{1},\limsup\limits_{n\rightarrow\infty}\frac{S_{n}(x,\beta)}{n}=\alpha_{2}\right\}, \alpha_1,\alpha_2\ge 0\)are obtained completely, where \(S_{n}(x,\beta)=\sum_{k=1}^{n}\varepsilon_k(x,\beta) \) denotes the sum of the first \(n\) digits of the \(\beta\)-expansion of \(x\). As an application, we present another concise proof of that the set of points \(x\in [0,1)\) satisfying \(\lim\limits_{n\to\infty}\frac{S_n(x,\beta)}{n}\) does not exist is of full Hausdorff dimension.
(21) B. Li, T. Sahlsten and T. Samuel, Intermediate beta-shifts of finite type, Discrete Contin. Dyn. Syst., 36(2016), no. 1, 323-344.[摘要] [链接]
An aim of this article is to highlight dynamical dierences betweenthe greedy, and hence the lazy, \(\beta\)-shift (transformation) and an intermediate \(\beta\)-shift (transformation), for a fixed \( \beta \in (1; 2)\). Specically, a classification in terms of the kneading invariants of the linear maps \(T_{\beta, \alpha} : x\mapsto \beta x+\alpha\ \text{mod}\ 1\)for which the corresponding intermediate \(\beta\)-shift is of fnite type is given. This characterisation is then employed to construct a class of pairs \((\beta, \alpha)\) such that the intermediate \(\beta\)-shift associated with \(T_{\beta,\alpha}\) is a subshift of fnite type. It is also proved that these maps \(T_{\beta, \alpha}\) are not transitive. This is in contrast to the situation for the corresponding greedy and lazy \(\beta\)-shifts and \(\beta\)-transformations,for which both of the two properties do not hold.
(20) L.-L. Fang, M. Wu, N.-R. Shieh and B. Li, Random Continued fractions: Levy constant and Chernoff-type estimate, J. Math. Anal. Appl., 429(2015), no. 1, 513-531.[摘要] [链接]
Given a stochastic process \(\{A_{n}, n ≥1\} \) taking values in natural numbers, the random continued fractions aredefined as \([A_1, A_2, ···, A_n, ···]\) analogously to the continued fraction expansion of real numbers. Assume that\(\{A_n, n≥1\}\) is ergodic and the expectation \(E(logA_{1})<\infty\), we give a Lévy-type metric theorem which covers that of real case presented by Lévy in 1929. Moreover, a corresponding Chernoff-type estimate is obtained under the conditions that\(\{A_n, n≥1\}\) is \(\psi\)-mixing and for each \( 0 <t <1\), \(E(A_t^1) <\infty\).
(19) M.-M. Kong, W.-C. Cheng and B. Li, Topological pressure for non autonomous systems, 76(2015), 82-92, Chaos Solitons Fractals,, 76(2015), 82-92,[摘要] [链接]
Kolyada and Snoha (1996) proposed the notion of entropy-like invariants for nonautonomousdynamical systems. This paper was based on that concept and involved extendingthe behavior of topological pressure to a fixed sequence of maps. Specifically, this studyinvestigated how the pressure changes when the potentials or the mappings vary. The analoguesof basic properties were obtained, and this study also reveals that, for any continuousmaps \(T\) and \(S\) from a compact metric space into itself, the maps \(T \circ S\) and\(S \circ T\) have thesame topological pressure (with respect to the corresponding potential functions).
(18) C.-H. Chen, H. Koivusalo, B. Li and V. Suomala, Projections of random covering sets, J. Fractal Geom. J. Fractal Geom., 1 (2014), no. 4, 449–467,[摘要] [链接]
We show that, almost surely, the Hausdorff dimension \(s_0 \) of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension \(k>s_0\). The result holds for random covering sets with a generating sequence of ball-like sets, and is obtained by investigating orthogonal projections of a class of random Cantor sets.
(17) J.-C. Ban and B. Li, The multifractal spectra for the recurrence rates of beta-transformations, J. Math. Anal. Appl., 420(2014), No. 2, 1662-1679.[摘要] [链接]
In this paper, we show a handy approximate approach to provide a lower bound of the Hausdorff dimension of a given subset in\([0, 1) \) related to β-transformation dynamical system. Here approximation means from special class with \(\beta\) shift satisfying the specification property or being subshift of finite type to general \(\beta >1\). As an application, we obtain the multifractal spectra for the recurrence rate of the first return time of \(\beta\) -transformation, including the cases returning to the ball and cylinder.
(16) R. Kuang, W.-C. Cheng, D.-K. Ma and B. Li, Different forms of entropy dimension for zero entropy systems, Dyn. Syst., 29(2014), No.2, 239-254.[摘要] [链接]
The aim of this paper is to introduce the lower \(s\)-topological entropy to distinguish zero entropy systems. This quantity is an invariant factor under topological conjugacy and power rule can be shown. Some examples are given to show that the lower entropy dimension can attain any value in \((0,1)\), and are different with the upper one and the entropy dimension in the sense of Bowen. A counterexample is used to indicate that the product rule does not hold, and the lower \(s\)-topological entropy of the subsystem for the non-wandering set can be strictly less than that of the system when \(0<s<1\). Finally, this study also constructs a dynamical system to show that the transitive system with zero entropy dimension may not be minimal.
(15) B. Li and V. Suomala, A note on the hitting probabilities of random covering sets, Ann. Acad. Sci. Fenn. Math., 39(2014), 625-633.[摘要] [链接]
Let \(E=\limsup\limits_{n\to\infty}(g_n+\xi_n)\) be the random covering set on the torus \(\mathbb{T}^d\), where \(\{g_n\}\) is a sequence of ball-like sets and \(\xi_n\) is a sequence of i.i.d. random variables uniformly distributed on \(T^d \). We prove that \(E\cap F\neq\emptyset\) almost surely whenever the Hausdorff dimension of the analytic set \(F\), \(\dim_H(F)>d-\alpha\), where \(\alpha\) is the almost sure Hausdorff dimension of \(E\). Moreover, examples are given to show that the condition on \(\dim_H(F) \) can not be replaced by the packing dimension of \(F\).
(14) B. Li, T. Persson, B. Wang and J. Wu, Diophantine approximation of the orbit of 1 in the dynamical system of beta expansions, Math. Z, 176(2014), 799-827.[摘要] [链接]
We consider the distribution of the orbits of the number 1 under the \(\beta\)-transformations \(T_\beta\) as \(\beta\) varies. Mainly, the size of the set of \(\beta > 1\) for which a given point can be well approximated by the orbit of 1 is measured by its Hausdorff dimension.The dimension of the following set \( E(\{\ell_n\}_{n\geq 1},x_0):=\{\beta>1:|T^n_\beta (1)-x_0|<\beta^{-\ell_n} \text{for infinitely many}\ n\in\Bbb N\} \) is determined, where \(x_0\) is a given point in \([0, 1]\) and \({\ell_n}n≥1\) is a sequence of integerstending to infinity as \(n →∞\). For the proof of this result, the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders (the set of \(\beta\) with a common prefix in the expansion of 1) in the parameter space \(\{\beta \in \mathbb{R} : \beta > 1 \}\).
(13) B. Li, B.-W. Wang, J. Wu and J. Xu, The shrinking target problem in the dynamical system of continued fractions, Proc. London Math. Soc. (3), 108(2014),159-186.[摘要] [链接]
Let \(([0, 1), T)\) be the dynamical system of continued fractions. Let\(\{z_n\}_{n \geq 1}\) be a sequenceof real numbers in \([0, 1]\) and \(\psi: \mathbb{N} \times [0, 1) → \mathbb{R}_+\) be a positive function. A point \(x \in [0, 1)\) is said to be \(\psi\)-approximable by \(\{z_n\}_{n \geq1}\) if \(|T_{n} x − z_n| \) < ψ(n, x) holds for infinitely many \(n \in \mathbb{N}\). In this paper, the Hausdorff dimension of the set of \(\psi\)-approximable points is studied. The dimensions are completely determined when \(\psi(n, x)=\psi(n)\) independent on \(x\) and when \(\psi(n, x) = e^{-(f(x)+···+f(T^{n−1}x)} \) with \( f \) a positive continuous function. For the proof of these results, a relationship between a ball in\([0, 1)\) and the cylinders defined by the partial quotientsin continued fractions is investigated. It is shown that a ball can be sufficiently packed by cylinders of the same order and of comparable length, which gives us explicit continued fraction representations in locating the points in a ball in \([0, 1)\).
(12) B. Li, N.-R. Shieh and Y.-M. Xiao, Hitting probabilities of the random covering sets, Fractal geometry and dynamical systems in pure and applied mathematics. II. Fractals in applied mathematics, 307-323, Contemp. Math. 601, Amer. Math. Soc., Providence, RI, 2013. Contemp. Math. 601, Amer. Math. Soc., Providence, RI. (3), 2013.[摘要] [链接]
Let \(E\) be the Dvoretzky random covering sets on the circle. By applyingthe method of limsup type random fractals, asillustrated in Khoshnevisan, Peres and Xiao, wedetermine the hitting probability \(P(E\cap G\ne \emptyset)\) and thepacking dimension of the intersection \(E \cap G\), where \(G\) is anarbitrary Borel set on the circle.
(11) X.-H. Hu, B. Li and J. Xu, Metric theorem and Hausdorff dimension on recurrence rate of Laurent series, Bull. Korean Math.(3), 51(2014), No.1, 157-171[摘要] [链接]
We show that the recurrence rates of Laurent series aboutcontinued fractions almost surely coincide with their pointwise dimensionsof the Haar measure. Moreover, let \(E_{\alpha,\beta} \) be the set of points with lowerand upper recurrence rates \(\alpha,\beta(0 \leq \alpha\leq \beta \leq \infty)\), we prove that all thesets\(E_{\alpha,\beta} \) are of full Hausdorff dimension. Then the recurrence sets \(E_{\alpha,\beta} \) have constant multifractal spectra.
(10) E. Järvenpää, M. Järvenpää, H. Koivusalo, B. Li and V. Hausdorff dimension of affine random covering sets in torus, Ann. Ann. Inst. Henri Poincare Probab. Stat(3), 50(2014) no.4, 1371-1384.[摘要] [链接]
We calculate the almost sure Hausdorff dimension of the random covering set \( lim sup_{n→∞}(g_n + ξ_n) \)in \(d\)-dimensional torus \( T^d\), where the sets \(g_n \subset T^d \) are parallelepipeds, or more generally, linear images of a set with nonempty interior, and \(ξ_n\in T_d\)are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linearmappings, holds provided that the sequences of the singular values are decreasing.
(9) W.-C. Cheng and B. Li,Topological pressure dimension, Chaos Solitons Fractals, 53(2013), 10-17.[摘要] [链接]
This paper presents the properties of topological pressure dimension, which is an extensionof the entropy dimension. Specifically, this paper studies the relationships among differenttypes of topological pressure dimension and identifies an inequality relating them. Thisanalysis calculates analogs of many known results of topological pressure. In particular,we will show the value of the pressure dimension is always equal to or greater than 1for any positive constant potential function.
(8) R. Kuang, W.-C. Cheng and B. Li,Fractal entropy of nonautonomous systems, Pacific J. Math, 262(2013), no. 2, 421-436.[摘要] [链接]
We define formulas of entropy dimension for a nonautonomous dynamicalsystem consisting of a sequence of continuous self-maps of a compact metricspace. This study reveals analogues of basic propositions for entropydimension, such as the power rule, product rule and commutativity, etc.These properties allow us to convert to an equality an inequality found byde Carvalho (1997) concerning the product rule for the autonomous dynamicalsystem. We also prove a subadditivity rule of entropy dimension forone-dimensional dynamics based on our previous work.
(7) D.-K. Ma, R. Kuang and B. Li, Topological entropy dimension for noncompact sets, Dyn. Syst, 262(2013), 27(2012), No. 3, 303-316.[摘要] [链接]
Similar with the fractal dimension, we introduce the concept of topologicalentropy dimension to classify the sets with entropy zero. We prove that theentropy dimension of the space in this article is not greater than that defined byDe Carvalho, where he introduced the entropy dimension for the system, and givesome examples indicating that such inequality is optimal. Some basic propositionsof entropy dimension are discussed and it turns out that the entropy dimension isinvariant under conjugacy. The property of the countable stability and a powerrule for the entropy dimension of any set are obtained. It is shown that any setshares the same entropy dimension with its image set.
(6) B. Li and Y.-C. Chen, Chaotic and topological properties of \(beta\)-transformations, J. Math. Anal. Appl. , 383(2011), 585-596.[摘要] [链接]
In this paper, we prove that the \(\beta\)-transformations are chaotic in the sense of both Li–Yorke and Devaney. The topological and metric properties of the sets of points with denseor non-dense orbits are investigated. We also prove the result that the set of points withnon-dense orbits under the \(\beta\)-transformation is of full Hausdorff dimension for any \(\beta>1.\)
(5) A. H. Fan, T. Langlet and B. Li, Quantitative uniform hitting in exponentially mixing systems, Recent developments in fractals and related fields, Appl. Numer. Harmon. Anal, Birkhäuser Boston, Inc,, Boston, MA, 2010[摘要] [链接]
Consider an exponentially mixing metric measure preserving system \((X,B, \mu, T,d)\). Let \(\alpha_{\max}\) be the maximal local dimension of \(\mu\). It is proved that if \(\tau<1/\alpha_{\max}\), then for \(\mu\)-almost all \(x\) and for every \(y\in X\) we have \(\liminf_{n\to \infty} n^\tau d(T^n x, y)=0\). The critical value \(1/\alpha_{\max}\) is optimal in manycase.
(4) W.-C. Cheng and B. Li,Zero entropy systems, J. Stat. Phys. , 140(2010), no. 5, 1006–1021.[摘要] [链接]
This paper introduces the notion of entropy dimension to measure the complexityof zero entropy dynamical systems, including the probabilistic and the topological versions.These notions are isomorphism invariants for measure-preserving transformation and continuity.We discuss basic propositions for entropy dimension and construct some examples toshow that the topological entropy dimension attains any value between 0 and 1. This paperalso gives a symbolic subspace to achieve zero topological entropy, but with full entropydimension.
(3) B. Li and J. Wu, Beta-expansion and continued fraction expansion over formal Laurent series, Finite Fields Appl , 14(2008), no. 3, 635–647[摘要] [链接]
Let \(x\in I \) be an irrational element and \(n\geq 1\), where \(I\) is the unit disc in the field of formal Laurent series \(F((X^{-1}))\), we denote by \(k_n(x)\) the number of exact partial quotients incontinued fraction expansion of \(x\), given by the first \(n\) digitsin the \(\beta\)-expansion of \(x\), both expansions are based on\(F((X^{-1}))\). We obtain that\(\liminf_{n\to+\infty}{\frac{k_n(x)}{n}}=\frac{\deg \beta}{2Q^\ast(x)}, \limsup_{n\to +\infty}{\frac{k_n(x)}{n}}=\frac{\deg\beta}{2Q_\ast(x)},\) where \(Q^\ast(x), Q_\ast(x)\) are the upper andlower constants of \(x\) respectively. Also, a central limit theoremand an iterated logarithm law for \(\{k_n(x)\}_{n\geq 1}\) areestablished.
(2) B. Li, J. Wu and J. Xu,Metric properties and exceptional sets of beta-expansions over formal series, Monatsh. Math, 155(2008), no. 2, 145–160[摘要] [链接]
This paper is concerned with the metric properties of \(\beta\)-expansions over the field of formal Laurent series. We will see that there are essential differences between \(\beta\)-expansions of the formal Laurent series case and the classical real case. Also the Hausdorff dimensions of some exceptional sets, with respect to the Haar measure, are determined.
(1) B. Li and J. Wu, Beta-expansion and continued fraction expansion, J. Math. Anal. Appl, 339(2008), no. 2, 1322–1331. [摘要] [链接]
For any real number \(\beta>1\), let \(\varepsilon(1, \beta)=\big(\varepsilon_{1}(1),\varepsilon_{2}(1), \cdots, \varepsilon_{n}(1), \cdots\big)\) be theinfinite \(\beta \)-expansion of 1. Define \(l_{n}=\sup\{k\geq0:\varepsilon_{n+j}(1)=0,\) for all \( 1\leq j\leq k\}\). Let\(x\in[0,1)\) be an irrational number. We denote by \(k_n(x)\) theexact number of partial quotients in continued fraction expansion of\(x\) given by the first \(n\) digits in the \(\beta\)-expansion of \(x\).If \(\limsup\limits_{n\to +\infty}{l_n}\) is finite, we obtain thatfor all \({x\in [0, 1)\backslash \mathbb{Q}}\),\(\liminf_{n\to+\infty}{\frac{k_n(x)}{n}}=\frac{\log\beta}{2\beta^\ast(x)}, \limsup_{n\to+\infty}{\frac{k_n(x)}{n}}=\frac{\log\beta}{2\beta_\ast(x)},\) where \(\beta^\ast(x)\), \(\beta_\ast(x)\) are the upper and lower Levy constants, which generalize the result in[ J. Wu, Continued fraction and decimalexpansions of an irrational number, Adv. Math. 206 (2) (2006) 684–694 ].Moreover, if \(\limsup\limits_{n\to+\infty}{\frac{l_n}{n}}=0\), we also get the similar result except a small set.
(55) 2017年11月10日 Hausdorff dimensions and hitting probabilities of random covering sets，庞加莱研究所，巴黎，法国
(54) 2017年9月28日 Hausdorff dimensions and hitting probabilities of random covering sets，Mittag-Leffler研究所，斯德哥尔摩，瑞典
(53) 2017年2月29日 Hausdorff dimensions of self-affine sets and nonconformal repellers，香港中文大学，香港
(52) 2016年11月1日 Chaotic and topological properties of continued fractions, 韩国高等科学院（KIAS），首尔，韩国
(51) 2016年10月28-30日 The dimension drop problem for self-affine sets, 第五届中南五省数学年会，南华大学，衡阳
(50) 2016年10月21-25日 Thermodynamic formalism for the singular value function，2016年全国动力系统学术研讨会，广西财经学院，南宁
(49) 2016年9月13日 Zero-one law of Hausdorff dimensions of the recurrent sets，淡江大学，台湾
(48) 2016年7月31日—8月2日 A Bernoulli-type measure related to beta-expansions, 2016年分形理论及相关问题学术研讨会—中国分形30年－回顾与展望，华中科技大学，武汉
(47) 2016年6月20-29日 Zero-one law of Hausdorff dimension of the recurrent sets，2016 Summer School on Fractal Geometry and Complex Dimensions, California Polytechinic State University, San Luis Obispo, 美国
(46) 2015年11月21-24日 The shrinking target problem in dynamical system of continued fractions, 中国数学会第十二次代表大会暨80周年纪念学术会议，首都师范大学，北京
(45) 2015年9月19-25日 Hitting probabilities of random covering sets in higher dimension, Fractals and related fields III. Porquerolles岛,法国
(44) 2015年9月18日Random affine code tree fractals and Falconer-Sloan condition,Seminar,庞加莱研究所，巴黎，法国
(43) 2015年8月24日 Some topics on random covering sets over the torus, Colloquium, 奥卢大学，奥卢，芬兰
(42) 2015年6月10-12日 Diophantine approximation in parameter spaces of the dynamical system of beta-transofrmations, Fractas and Numeration, Admont,奥地利
(41) 2015年3月30日 Some topics on topological entropy dimensions,Seminar， 香港中文大学，香港
(40) 2015年3月14—15日 Hitting probabilities of random covering sets in high dimension，2015 AMS Central Section Meeting－Special Section on Fractal and Tilings，Michigan State University，美国
(39) 2014年12月20—21日 Random affine code tree fractals and Falconer-Sloan condition，分形几何与相关专题研讨会，中山大学（珠海校区），珠海
(38) 2014年10月24—28日 Greedy, lazy and intermediate beta-expansions，Taiwan-Hong Kong-Mainland Workshop on Dynamics and Related Topics，华中师范大学，武汉
(37) 2014年10月17日 Dvoretzky random covering and potential theory,Seminar, 香港中文大学，香港
(36) 2014年8月8－12日 Intermediate beta-shifts of finite type， ICM2014 Satellite conference on dynamical systems and related topics, Chungnam National University, Daejeon, 韩国
(35) 2014年7月16日 Diophantine approximation in parameter spaces of the dynamical system of beta-transformations，Seminar, 中央研究院数学所，台北，台湾
(34) 2014年5月9-13日 中间有限型beta－移位，全国数学分形几何与动力系统会议，陕西师范大学，西安
(33) 2014年3月7日 The shrinking target problem and continued fractions，Seminar, 香港中文大学，香港
(32) 2013年12月20-21日 分形几何及相关问题研讨会，Random covering sets in high dimension,中山大学，广州
(31) 2013年12月16日，Some topics on the Hausdorff dimensions of the random covering sets, Seminar,华中科技大学，武汉
(30) 2013年6月11-14日 The shrinking problem in the dynamical system of continued fractions, Continued fractions, Interval exchanges and Applications to Geometry, Pisa大学, 意大利
(29) 2013年6月4日 The dimension theory of well-approximable sets and continued fractions, Colloquium, Bremen大学, 德国
(28) 2013年4月25日，The shrinking target problem in the dynamical system of continued fractions, 动力系统与遍历理论讨论班，Bristol大学，英国
(27) 2013年3月18-22日, Diophantine approximation of the orbit of 1 in the dynamical system of beta-expansions, Bremen Winter School on Multifractals and Number Theory, Bremen大学, 德国
(26) 2012年12月10-14日，Diophantine approximation of the orbit of 1 in beta-transformation dynamical system, International conference on advance on fractals and related topics，香港中文大学，香港
(25) 2012年10月4日，The distribution of the orbits in Beta-transformation dynamical system,Seminar, Helsinki大学数学系，Helsinki，芬兰
(24) 2012年4月23-27日，Hitting probabilities of the random covering sets, Ergodic methods in dynamics，Bedlewo数学研究与会议中心，波兰
(23) 2012年4月16-20日，The multifractal spectra for the recurrence rates of beta-transformations, Ergodic Theory and Dynamical Systems: Perspectives and Prospects，Warwick大学，英国
(22) 2012年2月16日，Random covering problems on the circle and for dynamical systems，Seminar, Helsinki大学数学系，Helsinki，芬兰
(21) 2011年9月12日，Some topics on random covering problem,Colloquium, Oulu大学数学系，Oulu,芬兰
(20) 2011年6月24日 Metric and topological properties of beta-expansionas,Seminar, 中央大学数学系，中坜，台湾
(19) 2011年6月13日 Some topics on entropy dimension,Seminar, 中央研究院数学所，台北，台湾
(18) 2011年5月27-31日 Some topics on beta-expansions, 全国分形几何与动力系统会议,张家界
(17) 2010年12月12-16日 Quantitative recurrence in exponentially mixing systems. 非线性分析国际会议,华南理工大学，广州
(16) 2010年11月26日 Some topics on entropy dimensions,Seminr, 中国科学院物理数学研究所，武汉
(15) 2010年8月20-23日 Random covering and recurrence problems. 复分析与分形几何暑期学校，湖南师范大学，长沙
(14) 2010年6月9日 Some dynamics related to the number theory and applications of Birkhoff ergodic theorem, Colloquium, 东海大学，台中，台湾
(13) 2010年5月12-15日 Recurrence and hitting problems of continued fraction and beta-shift dynamical systems. NCTS 2010 Workshop on Dynamical Systems, 清华大学，新竹，台湾
(12) 2010年5月4日 Quantitative recurrence and uniform hitting of dynamical systems. Colloquium, 东华大学，花莲，台湾
(11) 2010年4月23日 An introduction to \(\beta\)-expansion and symbolic dynamics III, Seminar, 中央研究院数学所，台北，台湾
(10) 2010年4月9日 An introduction to \(\beta\)-expansion and symbolic dynamics II, Seminar, 中央研究院数学所，台北，台湾
(9) 2010年3月26日 An introduction to \(\beta\)-expansion and symbolic dynamics I, Seminar, 中央研究院数学所，台北，台湾
(8) 2010年3月24日 Recurrence and hitting rate of a class of dynamical systems. Mini-Workshop on Applied Analysis and Probability, 台湾大学，台北，台湾
(7) 2010年3月19日 Recurrence and hitting problem in dynamical system, Seminar, 交通大学，新竹，台湾
(6) 2009年10月21-22日 Dynamical Systems Viewpoint of Continued Fraction and\(\beta\)-expansions. Workshop on Theory and Applications of Mathematical Analysis, 台湾大学，台北，台湾
(5) 2009年9月25日 Probability in Symbolic Dynamics. NCTS/TPE & TIMS Joint Activity in Probability, 台湾大学，台北，台湾
(4) 2009年9月23日 Some classical dynamical systems and applications of Birkhoff ergodic theorem, Colloquium, 辅仁大学，台北，台湾
(3) 2009年3月27日 A relationship between \(\beta\)-shift and continued fraction dynamical system. 厦门大学，厦门
(2) 2007年11月11日 Relationship between the beta-digits and the partial quotient of real numbers. Seminar on Probability and Ergodic Theory, LAMFA，Picardie大学，Amiens，法国
(1) 2007年6月23-25日 Beta-expansions and continued fraction expansion. 全国分形与动力系统会议，江苏大学，镇江
华南理工大学数学学院
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