Related papers: The Phi-dimension: A new homological measure

Hausdorff $\Phi$-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce an effective version of Hausdorff $\Phi$-dimension, which we call constructive $\Phi$-dimension. We prove a…

We consider one-parameter families of smooth uniformly contractive iterated function systems $\{f^\lambda_j\}$ on the real line. Given a family of parameter dependent measures $\{\mu_{\lambda}\}$ on the symbolic space, we study geometric…

Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax(\bmod 1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|\lambda_1|\le|\lambda_2|\le\dots\le|\lambda_d|$. We investigate the Hausdorff dimension of the…

On the space of isometric embeddings $f_g$ of metrics $g$ on a manifold $M^n$ into the standard $(\mb{S}^{\tn=\tn(n)},\tg)$, we consider the total exterior scalar curvature $\Theta_{f_g}(M)$, and squared $L^2$ norm of the mean curvature…

We develop a thermodynamic formalism for quasi-multiplicative potentials on a countable symbolic space and apply these results to the dimension theory of infinitely generated self-affine sets. The first application is a generalisation of…

Let $\psi:\mathbb R_+\to\mathbb R_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if the system $$|qx-p|< \, \psi(t) \ \ {\text{and}} \ \ |q|<t$$ has a non-trivial integer solution for all large…

The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this…

We show that a right artinian ring $R$ is right self-injective if and only if $\psi(M)=0$ (or equivalently $\phi(M)=0$) for all finitely generated right $R$-modules $M$, where $\psi, \phi : \mod R \to \mathbb N$ are functions defined by…

Let \fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\fa}(M), the finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M), the…

The (co)homological dimension of homomorphism $\phi:G\to H$ is the maximal number $k$ such that the induced homomorphism is nonzero for some $H$-module. The following theorems are proven: THEOREM 1. For every homomorphism $\phi:G\to H$ of a…

For a decreasing real valued function $\psi$, a pair $(A,\mathbf{b})$ of a real $m\times n$ matrix $A$ and $\mathbf{b}\in\mathbb{R}^m$ is said to be $\psi$-Dirichlet improvable if the system $$\|A\mathbf{q}+\mathbf{b}-\mathbf{p}\|^m <…

The dynamically defined measure (DDM) $\Phi$ arising from a finite measure $\phi_0$ on an initial $\sigma$-algebra on a set and an invertible map acting on the latter is considered. Several lower bounds for it are obtained and sufficient…

We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin.…

Given a finite set $\mathcal{A} \subseteq \mathrm{SL}(2,\mathbb{R})$ we study the dimension of the attractor $K_\mathcal{A}$ of the iterated function system induced by the projective action of $\mathcal{A}$. In particular, we generalise a…

We study continuity and discontinuity properties of some popular measure-dimension mappings under some topologies on the space of probability measures in this work. We give examples to show that no continuity can be guaranteed under general…

Let $f: M\rightarrow M$ be a continuous map on a compact metric space $M$ equipped with a fixed metric $d$, and let $\tau$ be the topology on $M$ induced by $d$. First, we will establish some fundamental properties of the mean Hausdorff…

We study self-similar sets and measures on $\mathbb{R}^{d}$. Assuming that the defining iterated function system $\Phi$ does not preserve a proper affine subspace, we show that one of the following holds: (1) the dimension is equal to the…

Let $\nu$ be a self similar measure on $\mathbb{R}^d$, $d\geq 2$, and let $\pi$ be an orthogonal projection onto a $k$-dimensional subspace. We formulate a criterion on the action of the group generated by the orthogonal parts of the IFS on…

We study invariant measures and thermodynamic formalism for a class of endomorphisms $F_T$ which are only piecewise differentiable on countably many pieces and non-conformal. The endomorphism $F_T$ has parametrized countably generated limit…

In 1996, Strichartz introduced reverse iterated function systems (RIFS) $\mathcal{F}=\{f_i(x)=r_i x+b_i\}_{i=1}^m$ of expanding mappings on $\mathbb{Z}$ and left the determination of the general dimension formulas of invariant sets as an…