English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

A well-known theorem of J.E. Hutchinson states that if an iterated functionsystem consists of similarity transformations and satisfies the open setcondition then its attractor supports a self-similar measure with Hausdorffdimension equal to the similarity dimension. In this article we prove thefollowing result which may be regarded as a form of partial converse: if aniterated function system consists of invertible affine transformations whoselinear parts do not preserve a common invariant subspace, and its attractorsupports a self-affine measure with Hausdorff dimension equal to the affinitydimension, then the system necessarily consists of similarity transformations.We obtain this result by showing that the equilibrium measures of an affineiterated function system are never Bernoulli measures unless the system eitheris reducible or consists of similarity transformations. The proof builds onearlier results in the thermodynamic formalism of affine iterated functionsystems due to Bochi, Feng, K\"aenm\"aki, Shmerkin and the first named authorand also relies on the work of Benoist on the spectral properties ofZariski-dense subsemigroups of reductive linear groups.

A converse statement to Hutchinson's theorem and a dimension gap for self-affine measures