Particle-size distribution (PSD) is a fundamental soil property usually reported as discrete clay, silt, and sand percentages. Models and methods to effectively generate a continuous PSD from such poor descriptions using another property would be extremely useful to predict and understand in fragmented distributions, which are ubiquitous in nature. Power laws for soil PSDs imply scale invariance (or selfsimilarity), a property which has proven useful in PSD description. This work is based on two novel ideas in modeling PSDs: (1) the concept of selfsimilarity in PSDs; and (2) mathematical tools to calculate fractal distributions for specific soil PSDs using few actual texture data. Based on these ideas, a random, multiplicative cascade model was developed that relies on a regularity of scale invariance called ‘log-selfsimilarity.’ The model allows the estimation of intermediate particle size values from common texture data. Using equivalent inputs, this new modeling approach was checked using soil data and shown to provide greatly improved results in comparison to the selfsimilar model for soil PSD data. The Kolmogorov-Smirnov D-statistic for the log-selfsimilar model was smaller than the selfsimilar model in 92.94% of cases. The average error was 0.74 times that of the selfsimilar model. The proposed method allows measurement of a heterogeneity index, H, defined using Hölder exponents, which facilitates quantitative characterization of soil textural classes. The average H value ranged from 0.381 for silt texture to 0.838 for sandy loam texture, with a variance of <0.034 for all textural classes. The index can also be used to distinguish textures within the same textural class. These results strongly suggest that the model and its parameters might be useful in estimating other soil physical properties and in developing new soil PSD pedotransfer functions. This modeling approach, along with its potential applications, might be extended to fine-grained mineral and material studies.

We give a $C^1$-perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/heteroclinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called a Markov iterated function system, a two-dimensional iterated function system in which the compositions are chosen in a Markovian way. Then we apply the result to the setting of three-dimensional partially hyperbolic diffeomorphisms.

This is the first paper in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for $C^1$ expanding maps on Riemannian manifolds.

Given a self-similar set K defined from an iterated function system $\Gamma =(\gamma _{1},\ldots ,\gamma _{d})$ and a set of functions $H=\{h_{i}:K\to \mathbb {R}\}_{i=1}^{d}$ satisfying suitable conditions, we define a generalized gauge action on Kajiwara–Watatani algebras $\mathcal {O}_{\Gamma }$ and their Toeplitz extensions $\mathcal {T}_{\Gamma }$ . We then characterize the KMS states for this action. For each $\beta \in (0,\infty )$ , there is a Ruelle operator $\mathcal {L}_{H,\beta }$ , and the existence of KMS states at inverse temperature $\beta $ is related to this operator. The critical inverse temperature $\beta _{c}$ is such that $\mathcal {L}_{H,\beta _{c}}$ has spectral radius 1. If $\beta <\beta _{c}$ , there are no KMS states on $\mathcal {O}_{\Gamma }$ and $\mathcal {T}_{\Gamma }$ ; if $\beta =\beta _{c}$ , there is a unique KMS state on $\mathcal {O}_{\Gamma }$ and $\mathcal {T}_{\Gamma }$ which is given by the eigenmeasure of $\mathcal {L}_{H,\beta _{c}}$ ; and if $\beta>\beta _{c}$ , including $\beta =\infty $ , the extreme points of the set of KMS states on $\mathcal {T}_{\Gamma }$ are parametrized by the elements of K and on $\mathcal {O}_{\Gamma }$ by the set of branched points.

This is the second part of our study on the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on $\mathbb {R}^d$ . In the first part [D.-J. Feng and K. Simon. Dimension estimates for $C^1$ iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of the attractor of every $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of every ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parameterized families of $C^1$ IFSs, and show that if the GTC is satisfied, then the dimensions of the IFS attractor and of the ergodic invariant measures are given by these upper bounds for almost every (in an appropriate sense) parameter. Moreover, we verify the GTC for some parameterized families of $C^1$ IFSs on ${\Bbb R}^d$ .

The notion of recurrent fractal interpolation functions (RFIFs) was introduced by Barnsley et al. [‘Recurrent iterated function systems’, Constr. Approx. 5 (1989), 362–378]. Roughly speaking, the graph of an RFIF is the invariant set of a recurrent iterated function system on $\mathbb {R}^2$. We generalise the definition of RFIFs so that iterated functions in the recurrent system need not be contractive with respect to the first variable. We obtain the box dimensions of all self-affine RFIFs in this general setting.

We study fast approximation of integrals with respect to stationary probability measures associated to iterated function systems on the unit interval. We provide an algorithm for approximating the integrals under certain conditions on the iterated function system and on the function that is being integrated. We apply this technique to estimate Hausdorff moments, Wasserstein distances and Lyapunov exponents of stationary probability measures.

Bilinear fractal interpolation surfaces were introduced by Ruan and Xu in 2015. In this paper, we present the formula for their box dimension under certain constraint conditions.

The group of ${\mathcal{C}}^{1}$ -diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$ -smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

In this paper we discuss the continuity of the Hausdorff dimension of the invariant set of generalised graph-directed systems given by contractive infinitesimal similitudes on bounded complete metric spaces. We use the theory of positive linear operators to show that the Hausdorff dimension varies continuously with the functions defining the generalised graph-directed system under suitable assumptions.

In this paper, we present a general framework to construct fractal interpolation surfaces (FISs) on rectangular grids. Then we introduce bilinear FISs, which can be defined without any restriction on interpolation points and vertical scaling factors.

We follow the idea of generalising the notion of classical iterated function systems, as presented by Mihail and Miculescu. We give their deliberations a more general setting and, using this general approach, study the generic aspect of the problem of existence of an attractor of a function system.

We introduce a geometric condition of Bloch type which guarantees that a subset of a bounded convex domain in several complex variables is degenerate with respect to every iterated function system. Furthermore, we discuss the relations of such a Bloch-type condition with the analogous hyperbolic Lipschitz condition.

The paper extends the rigidity of the mixing expanding repellers theorem of D. Sullivan announced at the 1986 IMC. We show that, for a regular conformal, satisfying the ‘Open Set Condition’, iterated function system of countably many holomorphic contractions of an open connected subset of a complex plane, the Radon–Nikodym derivative dμ/dm has a real-analytic extension on an open neighbourhood of the limit set of this system, where m is the conformal measure and μ is the unique probability invariant measure equivalent with m. Next, we introduce the concept of nonlinearity for iterated function systems of countably many holomorphic contractions. Several necessary and sufficient conditions for nonlinearity are established. We prove the following rigidity result: If h, the topological conjugacy between two nonlinear systems F and G, transports the conformal measure mF to the equivalence class of the conformal measure mG, then h has a conformal extension on an open neighbourhood of the limit set of the system F. Finally, we prove that the hyperbolic system associated to a given parabolic system of countably many holomorphic contractions is nonlinear, which allows us to extend our rigidity result to the case of parabolic systems.