Lower Assouad Type Dimensions Of Fractal Sets

Fractal sets are objects which contain details at arbitrary small scales,arbitrary complex structures,possibly,a sense of self-imilarity.They are widely studied in Math-ematics,Physics and related fields.Various forms of dimensions are the main tools to study fractal sets.The Assouad dimension and lower Assouad dimension,different from the Hausdorff dimension or box dimensions which focus on the global structures of a set,are sensitive to the local structure of fractal sets.Loosely speaking,the Assouad dimen-sion describes the‘thickest’part of a set,while the lower Assouad dimension concerns about the’thinnest’ part.Both are playing an important part in the dimension theory of fractal geometry.Assouad dimension was introduced by Assouad in 1970s and became an importan-t concept in the study of embedding problems,quasi-conformal mappings and fractal geometry.The lower Assouad dimension,introduced by Larman in 1960s,is a natural dual to the well-studied Assouad dimension.Unlike the Assouad dimension,the lower Assouad dimension does not have monotonicity,finite stability or quasi-Lipschitz invari-ance,which brings some troubles for further study.To treat these peculiarities,Fraser-Yu introduced the lower Assouad spectrum dimLθ F in 2016,which is a continuous function of θ∈(0,1)and quasi-Lipschitz invariant.Moreover,the limit of dimLθ F as θ→1 ap-proaches to dimL F for some fractal sets including McMullen sets.However,the lower Assouad spectrum is still far from being satisfactory.For example,it fails to be mono-tonic for θ∈(0,1);we do not know whether the limit of dimLθ F exists as θ→1 and 0→0;is dimLθ F able to approach dimL F as θ→1?This thesis is trying to solve these problems and the main results are stated as follows:1)We introduced alternative definitions called quasi-lower Assouad spectrum(dimLθ F)and quasi-lower Assouad dimension(dimqL F)by loosing the restrictions in the definition of lower Assouad spectrum.We showed that the quasi-lower Assouad spectrum is monotonic with respect to θ∈(0,1),it takes value between lower Assouad dimension and lower box dimension and is quasi-Lipschitz invariant.These results yield that the limit of quasi-lower Assouad spectrum dimLθ F as θ→1 and θ→0always exist when F is a subset of doubling metric spaces.2)We studied the relationship between dimLθ F and dimLθ F for a uniformly perfect set F in doubling metric spaces and obtained a variational principle between them,that is,dimLθ F is the infimum of dimLθ’ F as 0<θ’ ≤θ for any θ∈(0,1).Based on this,we then showed that the limit of dimLθ F as θ→1 and θ→0 exist.Moreover,limθ→1 dimLθ F=dimqL F,which provides a more accessible definition of dimqL F.3)It is usually hard to compute the lower Assouad dimension of a fractal set directly.Based on the work of Fraser-Yu,we derived the dimension formulae or bounds for the lower Assouad dimension of some fractal sets:a)we first obtained a formula for the lower Assouad dimension of compact sets in doubling metric spaces and then used it to give an explicit dimension formula for the homogeneous sets and a non-trivial lower bound for the lower Assouad dimension of Moran sets;b)we got explicit formulae for lower Assouad dimension and lower Assouad spectrum of Moran cut-out sets;c)we obtained the value of lower Assouad dimension,lower Assouad spectrum,modified lower Assouad dimension and box dimension for the graph of the Thomae function.These results show that there exist examples such that the lower Assouad dimension cannot be achieved by dimLθ F or dimLθ F as θ→1,nor the lower box dimension or modified lower Assouad dimension cannot be achieved by dimLθ F or dimLθ F as θ→0.