Ergodic fractal measures and dimension conservation

By H. Furstenberg. Ergodic Theory and Dynamical Systems, 2008

137 citations recorded by the source corpus.

Subfields: Condensed Matter Physics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/89e75863299d00a21777f31d6f96fd883e151dff

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In Condensed Matter Physics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Dimension of invariant measures for affine iterated function systems (2019, background): “ Rd with rj > 0, average contracting with respect to an ergodic m ∈ Mσ(Σ). Then for any proper subspace W of Rd, m ◦ π−1 is dimension conserving with respect to P W. This generalizes the result in [24, 33]. To see it, let p = dimW and let v1,...,vd be an orthonormal basis of Rd such that span(v 1”
Assouad Dimension and Fractal Geometry (2020, methodology): “We briefly describe some of the results from [115] to highlight the connection with Assouad dimension.”
On the Assouad dimension of self-similar sets with overlaps (2014, methodology): “This approach has been pioneered by Mackay and Tyson [19], but also harks back to Furstenberg’s notion of microsets [11].”
On self-similar sets with overlaps and inverse theorems for entropy (2012, background): “(The statement about intersections implies the one about projections using [11].)”
Equidistribution from fractal measures (2013, methodology): “An important phenomenon concerning these maps is measure rigidity: a well-known conjecture of Furstenberg states that, if m 6∼ n, then the only probability measures jointly invariant under Tm and Tn are combinations of Lebesgue measure and atomic measures on rational points.”
Dynamics on fractals and fractal distributions (2010, methodology): “Fractal distributions, which we define here, generalize Zähle’s scale-invariant distributions [30] and are very closely related to Furstenberg’s CP-processes [12, 14].”
Sixty years of fractal projections (2014, methodology): “Their proof uses the CP-chains of Furstenberg [31, 32], see also [33], and has three main ingredients: the lower semicontinuity of the expected Hausdorff dimension of the projection of a measure with respect to its ‘micromeasures’, Marstrand’s projection theorem, and the invariance of the dimension ”
Local entropy averages and projections of fractal measures (2009, methodology): “For the reader’s convenience, we summarize our main notation and typo- graphical conventions in Table 1.”
WEAK SEPARATION CONDITION, ASSOUAD DIMENSION, AND FURSTENBERG HOMOGENEITY (2015, methodology): “CP-processes were introduced by Furstenberg [11], greatly developed by Hochman [12], and studied by Käenmäki, Sahlsten, and Shmerkin [19].”
A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets (2016, background): “Recently, a more systematic study of CPprocesses was initiated by Furstenberg [17], with further developments by Gavish [18], Hochman [20], Hochman and Shmerkin [22] and others.”
On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions (2016, methodology): “c) Using Furstenberg’s theory of CP-processes and galleries [21], it is possible to obtain a version of Corollary 8.”