Title：Intermetiate Dimensions and Random Fractals

Speaker：Prof. Yimin Xiao(Michigan State University)

Time: January,6-8,2022， AM：9：00--11:00

Tengxun Meeting：351-3390-1790, PD:2022

Inviter：Associate Prof. Rui Kuang

Abstract：

    The notion of intermediate dimensions was introduced recently by Falconer, Fraser and Kempton [5] in 2019 to provide a continuum of dimensions between Hausdorff and box-counting. Several authors have applied intermediate dimensions to study deterministic and random fractals (cf. [1] [2] [3]). We refer to [4] for a survey.

In our lectures, we plan to cover the following topics:

(i) Definition and basic properties of intermediate dimensions;

(ii) Mathematical tools for computing intermediate dimensions;

(iii) Examples of self-similar or self-affine fractals;

(iv) Apply intermediate dimensions to study random fractals determined by the sample functions of Markov processes such as stable Levy processes (cf. [7]) or Gaussian random fields such as fractional Brownian motion ([1] [6] [8]).

References

[1] S. Burrell. Dimensions of fractional Brownian images, arxiv: 2002.03659.

[2] S. Burrell, K. J. Falconer and J. Fraser. Projection theorems for intermediate dimensions, to appear, J. Fractal Geom., arxiv: 1907.07632.

[3] K.J. Falconer. A capacity approach to box and packing dimensions of projections of sets and exceptional directions, J. Fractal Geom. to appear.

[4] K.J. Falconer. Intermediate dimensions - a survey. arXiv:2011.04363

[5] K. J. Falconer, J. M. Fraser and T. Kempton. Intermetiate dimensions, Math. Zeit. 296 (2020), 813-830.

[6] Y. Xiao. Packing dimension of the image of fractional Brownian motion, Statist. Probab. Lett. 33 (1997), 379-387.

[7] Y. Xiao. Random fractals and Markov processes. In: Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, (Michel L. Lapidus and Machiel van Frankenhuijsen, editors), pp. 261-338, American Mathematical Society, 2004.

[8] Y. Xiao. Sample path properties of anisotropic Gaussian random fields. In: A Minicourse on Stochastic Partial Differential Equations, (D. Khoshnevisan and F. Rassoul-Agha, editors), Lecture Notes in Math. 1962, pp. 145-212. Springer, New York, 2009.