6533b7d9fe1ef96bd126c33f
RESEARCH PRODUCT
Dynamics of the scenery flow and geometry of measures
Pablo ShmerkinAntti KäenmäkiTuomas Sahlstensubject
Pure mathematicsgeometryMatemáticasGeneral MathematicsDimension (graph theory)CONICAL DENSITIESDynamical Systems (math.DS)Measure (mathematics)Matemática Pura//purl.org/becyt/ford/1 [https]RECITFIABILITYEuclidean geometryClassical Analysis and ODEs (math.CA)FOS: MathematicsErgodic theoryscenery flowMathematics - Dynamical SystemsDIMENSIONMathematicsmatematiikkamathematicsta111measures//purl.org/becyt/ford/1.1 [https]Hausdorff spacePOROSITYConical surfacePrimary 28A80 Secondary 37A10 28A75 28A33Flow (mathematics)Mathematics - Classical Analysis and ODEsFRACTAL DISTRIBUTIONSDimension theorygeometriaCIENCIAS NATURALES Y EXACTASdescription
We employ the ergodic theoretic machinery of scenery flows to address classical geometric measure theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2015-03-26 |