The paper generalizes fractals to higher dimensions using affine transformations.
problem Generalizing fractals to higher dimensions.
method Using affine transformations and characterizations of affinely-equivalent Sierpinski carpet.
result Menger sponge and Sierpinski simplex in 4-dimensional space can be drawn out clearly.
The paper shows how certain groups' boundaries relate to the Sierpiński carpet.
problem Understanding the Bowditch boundaries of specific groups.
method Analyzing relatively hyperbolic groups and their boundaries.
result Groups with homeomorphic Bowditch boundaries to n-spheres are also relatively hyperbolic with n-1-dimensional Sierpiński carpet boundaries.
Uniform bounds found for Sierpinski carpet hyperbolic components.
problem Bounding hyperbolic components of Sierpinski carpet type.
method Establishing uniform a priori bounds and analyzing quadratic-like restrictions.
result Sierpinski carpet hyperbolic components of disjoint type are bounded.
A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…
We study a new class of square Sierpiński carpets Fn,p (5≤n,1≤p<2n−1) on S2, which are not quasisymmetrically equivalent to the standard Sierpiński carpets. We prove that the group of quasisymmetric self-maps of each Fn,p is the Euclidean isometry group. We also establish that …
The paper shows that a curl operator is not closable on carpet-like metric measure spaces.
problem The behavior of the curl operator on metric measure spaces, especially on carpet-like structures.
method Analyzing the exterior derivative and curl operators on 1-forms and 2-forms on metric measure spaces, particularly on Sierpinski carpets. result The curl operator is not closable on Sierpinski carpets and similar metric measure spaces.
We give a necessary and sufficient condition for a hyperbolic Coxeter group with planar nerve to have Sierpiński curve as its Gromov boundary.
If a torsion-free hyperbolic group G has 1-dimensional boundary, then the boundary is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When the boundary of G is a Sierpinski carpet we show that G is a quasi-convex subgroup of a 3-dimensional hyperbolic Poincare duality group. We also…
New groups with Menger curve boundaries found.
problem Finding non-hyperbolic groups with specific boundary shapes.
method Using Coxeter groups with complete graph nerves and embedding theorems.
result First non-hyperbolic CAT(0) groups with Menger curve boundaries discovered. Researchers create a Fredholm module on fractal shapes like the Cantor set.
problem Constructing Fredholm modules on complex fractal structures.
method Combining combinatorial techniques with higher-dimensional analogues.
result Calculated Dixmier trace of operators induced by the module.
The paper constructs thin Loewner carpets and their embeddings in S2.
problem Understanding the properties of Loewner carpets and their embeddings.
method Admissible quotiented inverse system construction for Loewner carpets and explicit embeddings.
result Explicit construction of infinitely many pairwise quasi-symmetrically distinct Q-Loewner carpets that admit quasisymmetric embeddings into S2. New subsets without interior support Poincaré inequalities, expanding previous results.
problem Finding subsets without interior that satisfy Poincaré inequalities.
method Employing uniform domains and measure density, focusing on boundary regularity and separation.
result Existence of subsets supporting Poincaré inequalities without interior, applicable to various spaces.
The paper examines properties of self-affine Sierpiński sponges using metric invariants.
problem Investigating properties of self-affine Sierpiński sponges using metric invariants.
method Examined through maximal power law property and perfectly disconnectedness.
result Characterized self-affine Sierpiński sponges by their metric properties.
New framework links fractal complexity to separation dimension.
problem Quantifying the complexity of fractal partitions.
method Introducing Separation Dimension ($\sepdim$) and Geometrically Regular Partitions (GRPs).
result Sharp upper bound for chromatic number of fractal partitions.
Proves convergence groups on a 2-sphere are Kleinian groups.
problem Proving convergence groups on a 2-sphere are Kleinian groups.
method Analyzing relatively hyperbolic groups with planar boundaries and applying to various versions of the Cannon conjecture.
result Proves relatively hyperbolic groups with planar boundaries are virtually Kleinian.
We study quasi-isometry invariants of Gromov hyperbolic spaces, focussing on the l_p-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous l_p-cohomology, thereby obtaining information about the l…
Characterizes Martin boundaries of certain hyperbolic groups.
problem Understanding the Martin boundaries of specific groups.
method Topological characterization using Green's function and random walks.
result Martin boundary matches CAT (0) boundary in some cases.
The paper studies degenerations of rational maps and their limits as geometrically finite rational maps.
problem Understanding the limits of quasi post-critically finite degenerations of rational maps.
method Constructing limits as geometrically finite rational maps on a tree of Riemann spheres, proving boundedness, and giving convergence criteria.
result Progress towards Thurston's compactness theorem and double limit theorem in complex dynamics.
The paper classifies boundaries of groups with isolated flats.
problem Classifying boundaries of groups with isolated flats.
method Classification theorem for 1-dimensional boundaries of groups with isolated flats.
result The boundary of a group with isolated flats is homeomorphic to a circle, Sierpinski carpet, or Menger curve.
New Coxeter groups yield n-dimensional Sierpiński boundaries.
problem Creating Coxeter groups with specific boundary shapes.
method Defined a class of right-angled Coxeter systems and provided conditions for their boundaries.
result Coxeter groups produce boundaries homeomorphic to n-dimensional Sierpiński compacta.
Paper defines topology automaton for Barański carpets and proves Hölder equivalence conditions.
problem Tackles Hölder equivalence of Barański carpets.
method Defines topology automaton and applies method from previous studies.
result Obtains sufficient condition for Hölder equivalence of Barański carpets.
Study shows how hyperbolic groups relate to Sierpinski spaces.
problem Understanding the relationship between hyperbolic groups and their boundary structures.
method Analyzes torsion-free hyperbolic groups and their visual boundaries, constructing examples and proving theorems.
result For n>6, hyperbolic groups with a specific boundary structure are equivalent to fundamental groups of certain manifolds.
Characterizes Coxeter groups with specific boundary shapes.
problem Identifying Coxeter groups with Sierpiński or Menger curve boundaries.
method Combining results from the literature on Gromov boundaries and Coxeter groups.
result Complete characterizations of hyperbolic Coxeter groups with Sierpiński or Menger curve boundaries.
Knots can be embedded into fractals like the Menger Sponge and Sierpinski Tetrahedron.
problem Embedding knots into fractals to compare complexity.
method Proved all knots can be embedded into Menger Sponge, and Pretzel knots into Sierpinski Tetrahedron. Compared iterations needed for given knots.
result Comparison of fractal complexity through knot embedding.
The paper extends Hodge-de Rham theory to higher-dimensional Sierpinski gaskets.
problem Analyzing differential forms and Laplacians on higher-dimensional fractal structures.
method Constructing sequences of graphs approximating Sierpinski gaskets, defining k-forms, de Rham derivatives, and their duals, proving harmonic properties, and exploring 2-forms.
result Obtained a basis for the space of harmonic 1-forms on level-3 Sierpinski gasket.
This paper constructs a continuous decomposition of the Sierpiński curve into acyclic continua one of which is an arc. This decomposition is then used to construct another continuous decomposition of the Sierpiński curve. The resulting decomposition space is homeomorphic to the continuum obtained from taking the Sierpi…
New non-rigid discrete groups found in hyperbolic spaces.
problem Uniqueness of conformal or spherical CR structures on spheres.
method Nilpotent Sierpiński carpet and stretching to construct non-rigid groups.
result Discrete hyperbolic groups can have non-rigid deformations.
Classifies horocycle flow closures in hyperbolic 3-manifolds.
problem Classifying horocycle flow closures in hyperbolic 3-manifolds.
method Classifies orbit closures of the 1-dimensional horocycle flow on the frame bundle of M.
result The closure of a horocycle in M is a properly immersed submanifold.
New energy forms on fractal curves solve geometric complexities.
problem Building a Laplacian on fractal domains with specific geometry.
method Introduced energy forms with normalization constants to account for fractal geometry.
result Same Laplacian for Sierpiński arrowhead curve and Sierpiński gasket.
This paper classifies fundamental groups of perforated surfaces.
problem Classifying the fundamental groups of perforated surfaces.
method Using the classification theorem for surfaces and covering spaces.
result Fundamental groups of perforated surfaces are large and not Hopfian.
We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic 3 manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schott…
A theorem controls the relationship between dimensions of continua.
problem Understanding the relationship between different dimensions of continua.
method Introduced a controlled version of the Hahn-Mazurkiewicz Theorem.
result Established a relationship between SDim(X) and HDim(X). We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian…
Estimates heat kernel gradients on fractal-like cable systems.
problem Bounding gradients of heat kernels on complex fractal structures.
method Pointwise upper estimates for heat kernel gradients.
result Derives Lp-boundedness of quasi-Riesz transforms. We start by a review of the chronology of mathematical results on the Dirichlet-to-Neumann map which paved the way towards the physics of transformational acoustics. We then rederive the expression for the (anisotropic) density and bulk modulus appearing in the pressure wave equation written in the transformed coordina…
A brief historical perspective is first given concerning financial crashes, - from the 17th till the 20th century. In modern times, it seems that log periodic oscillations are found before crashes in several financial indices. The same is found in sand pile avalanches on Sierpinski gaskets. A discussion pertains to the…
This is a simple mathematical introduction into Feynman diagram technique, which is a standard physical tool to write perturbative expansions of path integrals near a critical point of the action. I start from a rigorous treatment of a finite dimensional case (which actually belongs more to multivariable calculus than …
Paper defines new sets and calculates their Hausdorff dimensions.
problem Calculating Hausdorff dimensions of new self-similar sets.
method Introduced asymptotic self-similar sets and new geometric constructions.
result Determined Hausdorff dimensions of new sets.
Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either…
IFS with average shadowing property ensure chain recurrence and have specific examples.
problem Analyzing the average shadowing property in IFS.
method Examining uniformly contracting, conjugacy, and product IFS; proving chain recurrence; introducing examples.
result IFS with average shadowing property ensure chain recurrence.
We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of …
New subharmonicity concept proves conjecture on Riemannian manifolds.
problem Proving positivity of solutions to a specific PDE on Riemannian manifolds.
method Introducing local λ-shift defectivity and studying it on locally smoothing spaces. result Proof of Braverman, Milatovic, Shubin conjecture on positivity of solutions.
Study finds a measure for sponge components of Lalley-Gatzouras type.
problem Understanding the distribution of δ-connected components in self-affine sponges.
method Generalized existing results to self-affine sponges of Lalley-Gatzouras type, proving a measure relationship.
result Existence of a Bernoulli measure for cylinder components with a specific asymptotic relation.
Solvency II's V@R method hides downside risk, study shows.
problem Solvency II's V@R method fails to capture extreme risks.
method Analyzes distortion risk measures and network portfolio allocations.
result Firms can reduce capital requirements by transferring risk within a network.
New fractal spaces not quasisymmetric to Loewner spaces discovered.
problem Finding new fractal spaces not quasisymmetric to Loewner spaces.
method Introduced iterated graph systems (IGS) to create new fractal spaces.
result Disproved Kleiner's conjecture about self-similar fractals.
We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group Γ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit e…
This paper deals with both complex dynamical systems and conformal iterated function systems. We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a d-parameter family of such semigroups satisfies the transversality condition, then for almost every par…