0000000000448684

AUTHOR

Eino Rossi

showing 4 related works from this author

Local dimensions of measures on infinitely generated self-affine sets

2014

We show the existence of the local dimension of an invariant probability measure on an infinitely generated self-affine set, for almost all translations. This implies that an ergodic probability measure is exactly dimensional. Furthermore the local dimension equals the minimum of the local Lyapunov dimension and the dimension of the space. We also give an estimate, that holds for all translation vectors, with only assuming the affine maps to be contractive.

Discrete mathematicsmatematiikka28A80Applied Mathematicsta111Minkowski–Bouligand dimensionDimension functionMetric Geometry (math.MG)Dynamical Systems (math.DS)Complex dimensionEffective dimensionPacking dimensionMathematics - Metric GeometryHausdorff dimensionFOS: MathematicsdimensionsMathematics - Dynamical SystemsDimension theory (algebra)Inductive dimensionulottuvuudetAnalysisMathematicsJournal of Mathematical Analysis and Applications
researchProduct

Self-affine sets with fibered tangents

2016

We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then for each generic point there exists a rotation $\mathcal O$ such that all tangent sets at that point are either of the form $\mathcal O((\mathbb R \times C) \cap B(0,1))$, where $C$ is a closed porous set, or of the form $\mathcal O((\ell \times \{ 0 \}) \cap B(0,1))$, where $\ell$ is an interval.

Pure mathematicsClass (set theory)General MathematicsDynamical Systems (math.DS)Interval (mathematics)iterated function system01 natural sciencesself-affine setGeneric pointLine segmentstrictly self-affine sets0103 physical sciencesClassical Analysis and ODEs (math.CA)FOS: MathematicsPoint (geometry)Porous set0101 mathematicsMathematics - Dynamical SystemsMathematicsApplied Mathematics010102 general mathematicsta111Tangenttangent setsTangent setMathematics - Classical Analysis and ODEs010307 mathematical physicsAffine transformation
researchProduct

Rigidity of quasisymmetric mappings on self-affine carpets

2016

We show that the class of quasisymmetric maps between horizontal self-affine carpets is rigid. Such maps can only exist when the dimensions of the carpets coincide, and in this case, the quasisymmetric maps are quasi-Lipschitz. We also show that horizontal self-affine carpets are minimal for the conformal Assouad dimension.

Class (set theory)Pure mathematicsMathematics::Dynamical SystemsGeneral Mathematicsquasisymmetric mapsMathematics::General TopologyPhysics::OpticsConformal mapRigidity (psychology)01 natural sciencesDimension (vector space)0103 physical sciencesClassical Analysis and ODEs (math.CA)FOS: MathematicsMathematics::Metric Geometry0101 mathematicsself-affine carpetsMathematicsta111010102 general mathematicsPhysics::Classical PhysicsMathematics - Classical Analysis and ODEs010307 mathematical physicsAffine transformation28A80 37F35 30C62 30L10
researchProduct

Weak separation condition, Assouad dimension, and Furstenberg homogeneity

2015

We consider dimensional properties of limit sets of Moran constructions satisfying the finite clustering property. Just to name a few, such limit sets include self-conformal sets satisfying the weak separation condition and certain sub-self-affine sets. In addition to dimension results for the limit set, we manage to express the Assouad dimension of any closed subset of a self-conformal set by means of the Hausdorff dimension. As an interesting consequence of this, we show that a Furstenberg homogeneous self-similar set in the real line satisfies the weak separation condition. We also exhibit a self-similar set which satisfies the open set condition but fails to be Furstenberg homogeneous.

General MathematicsHomogeneity (statistics)ta111Open setPrimary 28A80 Secondary 37C45 28D05 28A50Moran constructioniterated function systemSet (abstract data type)CombinatoricsDimension (vector space)dimensionMathematics - Classical Analysis and ODEsweak separation conditionClassical Analysis and ODEs (math.CA)FOS: MathematicsLimit (mathematics)Limit setCluster analysisReal lineMathematics
researchProduct