Weak Separation Condition Research Articles

Separation Conditions for Iterated Function Systems with Overlaps on Riemannian Manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau et al. (Monatsch Math 156:325–355, 2009). We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin and Yau (Commun Anal Geom 13:821–843, 2005) and Lau and Ngai (Adv Math 208:647–671, 2007) on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai et al. (Nonlinearity 23:2333–2350, 2010).

On the structure of self-affine Jordan arcs in ℝ2

Abstract We prove that if a self-affine arc γ ∈ R 2 \gamma \in {{\mathbb{R}}}^{2} does not satisfy weak separation condition, then it is a segment of a parabola or a straight line. If a self-affine arc γ \gamma is not a segment of a parabola or a line, then it is a component of the attractor of a Jordan multizipper with the same set of generators.

The numerical prediction of the Earth’s polar motion based on an advanced multivariate algorithm

Since there are complicated changes in the polar motion (PM) from sub-annual to decadal, precisely predicting it is challenging. Here, we provide an advanced multivariate algorithm by combining an iterative oblique singular spectrum analysis (IOSSA) with pseudo data (IOSSApd) and consider more periodic and quasi-periodic signals, especially long-period oscillations (Ding et al., Geophys. Res. Lett., 2019, 46, 13765–13774) and multi-frequency Chandler wobble (Pan, International Journal of Geosciences, 2012, 3, 930–951), than previous studies. The IOSSA in oblique coordinates, due to its weak separability conditions, has a better separation performance than general singular spectrum analysis (SSA), and the IOSSApd approach further solved the shift problem. Upon using the IOSSApd method, the PM data can be separated into deterministic and stochastic components, extrapolated by the multiple-harmonic (MH) and autoregressive integrated moving average (ARIMA) models, respectively. Based on the IERS EOPC04 PM series, we produced multiple sets of PM predictions with a 1-year leading time and reported the IERS Bulletin A predictions as a comparison. For 90-day leading time predictions, the mean absolute errors (MAEs) of the x- and y-components were 7.69 and 5.12 mas, respectively, while the corresponding MAEs obtained by IERS Bulletin A were 9.45 and 5.69 mas, respectively. For up to 360 days, our algorithm obtains the MAEs of PM slowly accumulating to 12.98 mas on average, far better than the 19.14 mas for Bulletin A’s predictions (also significantly superior to the corresponding results given by previous studies). The prediction performance in the middle- and long-term prediction is further compared against the general SSA predictor. By virtue of weak periodic error, our results show that combining the IOSSApd + MH + ARIMA models improved the prediction success rate up to 75.39% and 69.58% for the x- and y-component, respectively.

Local dimensions of self-similar measures satisfying the finite neighbour condition* *KEH was supported by NSERC Grant 2016-03719. AR was supported by this grant as well as EPSRC Grant EP/V520123/1.

We study sets of local dimensions for self-similar measures in satisfying the finite neighbour condition, which is formally stronger than the weak separation condition (WSC) but satisfied in all known examples. Under a mild technical assumption, we establish that the set of attainable local dimensions is a finite union of (possibly singleton) compact intervals. The number of intervals is bounded above by the number of non-trivial maximal strongly connected components of a finite directed graph construction depending only on the governing iterated function system. We also explain how our results allow computations of the sets of local dimensions in many explicit cases. This contextualises and generalises a vast amount of prior work on sets of local dimensions for self-similar measures satisfying the WSC.

SEPARATION PROPERTIES FOR BI-LIPSCHITZ ITERATED FUNCTION SYSTEMS

In this paper, we study the iterated function systems (IFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focusing on IFSs that do not satisfy the open set condition. We define a weak* separation condition (W*SC), which is strictly weaker than the weak separation condition of Lau and Ngai. By assuming the bounded distortion property, we show that the W*SC is equivalent to the identity limit criterion of Zerner for such IFSs. In particular, in the one-dimensional case, we show that the W*SC is also equivalent to Ahlfors–David regularity, the Assouad dimension is strictly less than [Formula: see text] and positivity of the [Formula: see text]-dimensional Hausdorff measure, where [Formula: see text] is the zero of the topological pressure.

Geometric and combinatorial properties of self-similar multifractal measures

AbstractFor any self-similar measure $\mu $ in $\mathbb {R}$ , we show that the distribution of $\mu $ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the iterated function system of similarities (IFS). This generalizes the net interval construction of Feng from the equicontractive finite-type case. When the measure satisfies the weak separation condition, we prove that this directed graph has a unique attractor. This allows us to verify the multifractal formalism for restrictions of $\mu $ to certain compact subsets of $\mathbb {R}$ , determined by the directed graph. When the measure satisfies the generalized finite-type condition with respect to an open interval, the directed graph is finite and we prove that if the multifractal formalism fails at some $q\in \mathbb {R}$ , there must be a cycle with no vertices in the attractor. As a direct application, we verify the complete multifractal formalism for an uncountable family of IFSs with exact overlaps and without logarithmically commensurable contraction ratios.

On dimensions of visible parts of self-similar sets with finite rotation groups

We derive an upper bound for the Assouad dimension of visible parts of self-similar sets generated by iterated function systems with finite rotation groups and satisfying the weak separation condition. The bound is valid for all visible parts and it depends on the direction and the penetrable part of the set, which is a concept defined in this paper. As a corollary, we obtain in the planar case that if the projection is a finite or countable union of intervals then the visible part is 1-dimensional. We also prove that the Assouad dimension of a visible part is strictly smaller than the Hausdorff dimension of the set provided the projection contains interior points. Our proof relies on Furstenberg’s dimension conservation principle for self-similar sets.

The scenery flow of self-similar measures with weak separation condition

AbstractWe show that self-similar measures on$\mathbb R^d$satisfying the weak separation condition are uniformly scaling. Our approach combines elementary ergodic theory with geometric analysis of the structure given by the weak separation condition.

SELF-AFFINE SETS: THE RELATION BETWEEN POSITIVE LEBESGUE MEASURE AND NON-EMPTY INTERIOR

For the self-affine IFS [Formula: see text] satisfying the weak separation condition (WSC), let [Formula: see text] be the corresponding self-affine set. It is proven in this paper that [Formula: see text] has positive Lebesgue measure only if the interior of [Formula: see text] is non-empty.

When the weak separation condition implies the generalized finite type condition

We prove that an iterated function system of similarities on R \mathbb {R} that satisfies the weak separation condition and has an interval as its self-similar set is of generalized finite type. It is unknown if the assumption that the self-similar set is an interval is necessary.

On multifractal formalism for self-similar measures with overlaps

Let $$\mu $$ be a self-similar measure generated by an IFS $$\varPhi =\{\phi _i\}_{i=1}^\ell $$ of similarities on $${{\mathbb {R}}}^d$$ ( $$d\ge 1$$ ). When $$\varPhi $$ is dimensional regular (see Definition 1.1), we give an explicit formula for the $$L^q$$ -spectrum $$\tau _\mu (q)$$ of $$\mu $$ over [0, 1], and show that $$\tau _\mu $$ is differentiable over (0, 1] and the multifractal formalism holds for $$\mu $$ at any $$\alpha \in [\tau _\mu '(1),\tau _\mu '(0+)]$$ . We also verify the validity of the multifractal formalism of $$\mu $$ over $$[\tau _\mu '(\infty ),\tau _\mu '(0+)]$$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to the recent result of Shmerkin (Ann. Math. (2) 189(2):319–391, 2019) on the $$L^q$$ -spectrum of self-similar measures.

Quasi-doubling of self-similar measures with overlaps

The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which we call the Assouad dimension (of the measure) and is also known as the upper regularity dimension. One reason for the interest in this notion is that a measure has finite Assouad dimension if and only if it is doubling. Motivated by recent progress on both the Assouad dimension of measures that satisfy a strong separation condition and the quasi-Assouad dimension of metric spaces, we introduce the notion of the quasi-Assouad dimension of a measure. As with sets, the quasi-Assouad dimension of a measure is dominated by its Assouad dimension. It dominates both the quasi-Assouad dimension of its support and the supremal local dimension of the measure, with strict inequalities possible in all cases. Our main focus is on self-similar measures in \mathbb R whose support is an interval and which may have ‘overlaps’. For measures that satisfy a weaker condition than the weak separation condition we prove that finite quasi-Assouad dimension is equivalent to quasi-doubling of the measure, a strictly less restrictive property than doubling. Further, we exhibit a large class of such measures for which the quasi-Assouad dimension coincides with the maximum of the local dimension at the endpoints of the support. This class includes all regular, equicontractive self-similar measures satisfying the weak separation condition, such as convolutions of uniform Cantor measures with integer ratio of dissection. Other properties of this dimension are also established and many examples are given.

Dimensions in infinite iterated function systems consisting of bi-Lipschitz mappings

We study infinite iterated function systems (IIFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focussing on IFSs that do not satisfy the open set condition. By assuming the logarithmic distortion property and some cardinality growth condition, we obtain a formula for the Hausdorff, box, and packing dimensions of the limit set in terms of certain topological pressure. By assuming, in addition, the weak separation condition, we show that these dimensions are equal to the growth dimension of the limit set.

Self‐conformal sets with positive Hausdorff measure

We investigate the Hausdorff measure and content on a class of quasi self-similar sets that include, for example, graph-directed and sub self-similar and self-conformal sets. We show that any Hausdorff measurable subset of such a set has comparable Hausdorff measure and Hausdorff content. In particular, this proves that graph-directed and sub self-conformal sets with positive Hausdorff measure are Ahlfors regular, irrespective of separation conditions. When restricting to self-conformal subsets of the real line with Hausdorff dimension strictly less than one, we additionally show that the weak separation condition is equivalent to Ahlfors regularity and its failure implies full Assouad dimension. In fact, we resolve a self-conformal extension of the dimension drop conjecture for self-conformal sets with positive Hausdorff measure by showing that its Hausdorff dimension falls below the expected value if and only if there are exact overlaps.

Index Numbers and Revealed Preference Rankings

For previously identified weakly separable blockings of goods and assets, we construct aggregates using four superlative index numbers, the Fisher, Sato-Vartia, Tornqvist and Walsh, two non-superlative indexes, the Laspeyres and Paasche and the atheoretical simple summation. We conduct several tests to examine how well each of these aggregates “fit” the data. These tests are how close the aggregates come to solving the revealed preference conditions for weak separability, how often each aggregate gets the direction of change correct and how well the aggregates mimic the preference ranking from revealed preference tests. We find that, as the number of goods and assets being aggregated increases, the problems with simple summation manifest.

INDEX NUMBERS AND REVEALED PREFERENCE RANKINGS

For previously identified weakly separable blockings of goods and assets, we construct aggregates using four superlative index numbers, the Fisher, Sato-Vartia, Törnqvist, and Walsh, two non-superlative indexes, the Laspeyres and Paasche, and the atheoretical simple summation. We conduct several tests to examine how well each of these aggregates “fit” the data. These tests are how close the aggregates come to solving the revealed preference conditions for weak separability, how often each aggregate gets the direction of change correct, and how well the aggregates mimic the preference ranking from revealed preference tests. We find that, as the number of goods and assets being aggregated increases, the problems with simple summation manifest.

Variational principle and zero temperature limits of asymptotically (sub)-additive projection pressure

Let $$\{S_i\}_{i=1}^l$$ be an iterated function system (IFS) on ℝd with an attractor K. Let (∑, σ) denote the one-sided full shift over the finite alphabet {1, 2,...,l}, and let π : ∑ → K be the coding map. Given an asymptotically (sub)-additive sequence of continuous functions ℱ = {fn}n⩾1; we define the asymptotically additive projection pressure Pπ(ℱ) and show the variational principle for Pπ(ℱ) under certain affne IFS. We also obtain variational principle for the asymptotically sub-additive projection pressure if the IFS satisfies asymptotically weak separation condition (AWSC). Furthermore, when the IFS satisfies AWSC, we investigate the zero temperature limits of the asymptotically sub-additive projection pressure Pπ(βℱ) with positive parameter β.

Local dimensions of measures of finite type III — Measures that are not equicontractive

We extend the study of the multifractal analysis of the class of equicontractive self-similar measures of finite type to the non-equicontractive setting. Although stronger than the weak separation condition, the finite type property includes examples of IFS that fail the open set condition. The important combinatorial properties of equicontractive self-similar measures of finite type are extended to the non-equicontractive setting and we prove that many of the results from the equicontractive case carry over to this new, more general, setting. In particular, previously it was shown that if an equicontractive self-similar measure of finite type was regular, then the calculations of local dimensions were relatively easy. We modify this definition of regular to define measures to be generalized regular. This new definition will include the non-equicontractive case and we obtain similar results. Examples are studied of non-equicontractive self-similar generalized regular measures, as well as equicontractive self-similar measures which are generalized regular in this new sense, but which are not regular.

On hyperbolic graphs induced by iterated function systems

For any contractive iterated function system (IFS, including the Moran systems), we show that there is a natural hyperbolic graph on the symbolic space, which yields the Hölder equivalence of the hyperbolic boundary and the invariant set of the IFS. This completes the previous studies ([16,20,30]) by eliminating superfluous conditions, and admits more classes of sets (e.g., the Moran sets). We also show that the bounded degree property of the graph can be used to characterize certain separation properties of the IFS (open set condition, weak separation condition); the bounded degree property is particularly important when we consider random walks on such graphs. This application and the other application to Lipschitz equivalence of self-similar sets will be discussed.

The intersections of self-similar and self-affine sets with their perturbations under the weak separation condition

For a self-similar or self-affine iterated function system (IFS), let$\unicode[STIX]{x1D707}$be the self-similar or self-affine measure and$K$be the self-similar or self-affine set. Assume that the IFS satisfies the weak separation condition and$K$is totally disconnected; then, by using the technique of neighborhood decomposition, we prove that there is a neighborhood$\unicode[STIX]{x1D6FA}$of the identity map Id such that$\sup \{\unicode[STIX]{x1D707}(g(K)\cap K):g\in \unicode[STIX]{x1D6FA}\setminus \{\text{Id}\}\}<1$.