ABSTRACT
In the study of fractal sets M the notion of dimensions plays an important role in which the Hausdorff dimension denoted by HD(M), the upper and lower box dimensions denoted by C ¯ ( M ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq2157.tif"/> and C _ ( M ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq2158.tif"/> are considered most frequently. Similarily, for Borel probability measure μ on a compact metric space M we can study HD ( μ ) , C _ L ( μ ) , C ¯ L ( μ ) , C _ ( μ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq2159.tif"/> and C ¯ ( μ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq2160.tif"/> which are defined as follows (see Young [5] and Ledrappier [2]) HD ( μ ) = def inf Y ⊂ M μ Y = 1 HD ( Y ) ; C _ L ( μ ) = def sup δ → 0 lim inf ε → 0 log N ( ε , δ ) log ( 1 / ε ) ; C ¯ L ( μ ) = def sup δ → 0 lim sup ε → 0 log N ( ε , δ ) log ( 1 / ε ) ; C _ ( μ ) = def sup δ → 0 inf Y ⊂ M μ Y ⩾ 1 − δ C _ ( Y ) ; C ¯ ( μ ) = def sup δ → 0 inf Y ⊂ M μ Y ⩾ 1 − δ C ¯ ( Y ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003067498/6ba0c26b-f086-40cc-86ed-1693dc596221/content/eq2161.tif"/>
