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Purpose The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus. Design/methodology/approach The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited. Findings The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead. Originality/value For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$. These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family $f_{\lambda}$. Again there are similarities to and differences from the parameter plane of the family $P_a$ and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points.

The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar\'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has \emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.

We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel–Anosov–Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms with positive finite entropy. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds.We construct an example of a minimal, weakly mixing and uniformly rigid homeomorphism of the pseudo-circle, and by our method we are also able to extend it to other one-dimensional hereditarily indecomposable continua, thereby producing the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension 1. We also show that the examples we construct can be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of connected spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least 2.

Ovaj diplomski rad bavi se raznim analitičkim konstrukcijama Cantorovog skupa. U prvom poglavlju upoznajemo se s Cantorovim skupom i konstruiramo ga na tzv. geometrijski način te objašnjavamo neka njegova svojstva (duljinu, kardinalitet i gustoću u skupu \(\mathbb{R}\)). U drugom poglavlju bavimo se prvom konstrukcijom Cantorovog skupa preko tzv. verižnih razlomaka. Definiramo verižne razlomke, dokazujemo postojanje konačnog prikaza svakog racionalnog broja u obliku verižnog razlomka i postojanje razvoja u verižni razlomak za svaki realni broj, odnosno, dokazujemo konvergenciju beskonačnih verižnih razlomaka. Naposlijetku dokazujemo da je, za cijeli broj \(k\), \(k\geq2\), skup svih realnih brojeva \(\alpha\), takvih da je \(0\leq \alpha\leq k^{-1}\) i takvih da razvoj broja \(\alpha\), u verižni razlomak ne sadrži parcijalni kvocijent manji od k, Cantorov skup. U trećem poglavlju konstruiramo Cantorov skup kao homeomorfnu sliku prstena cijelih \(p\)-adskih brojeva \(\mathbb{Z}_p\), \(p\) prost broj. Prvo definiramo \(p\)-adske brojeve i cijele \(p\)-adske brojeve. Definiramo \(p\)-adsku normu i \(p\)-adsku udaljenost na polju \(\mathbb{Q}\) te dokazujemo da polje \(\mathbb{Q}\) s \(p\)-adskom normom nije potpuno. Upotpunjujemo polje \(\mathbb{Q}\) poljem klasa ekvivalencije Cauchyjevih nizova, \(\mathbb{Q}_p \). Nakon toga, pokazujemo ekvivalenciju polja \(\mathbb{Q}_p \) kao skupa klasa ekvivalencija nizova i beskonačnih \(p\)-adskih ekspanzija koje konvergiraju u \(p\)-adskoj normi na \(\mathbb{Q}_p \). Konačno, pokazujemo postojanje homeomorfizma izmedu prstena cijelih \(p\)-adskih brojeva \(\mathbb{Z}_p\) i Cantorovog skupa u naslijeđenoj euklidskoj topologiji. This thesis deals with various analytic constructions of the Cantor set. In the first chapter, we get acquainted with the Cantor set and construct it on the so-called geometric way and we explain some of its properties (length, cardinality and density in the set \(\mathbb{R}\)). In the second chapter, we deal with the first construction of the Cantor set through the so-called continued fractions. We define continued fractions, we prove the existence of a finite representation of every rational number in the form of a continued fraction and the existence of a development into a continued fraction for every real number, that is, we prove the convergence of infinite continued fractions. Finally, we prove that, for an integer \(k\), \(k\geq2\), the set of all real numbers \(\alpha\), such that \(0\leq \alpha\leq k^{-1 }\) and such that the development of the number \(\alpha\) into a continued fraction does not contain a partial quotient smaller than k, the Cantor set. In the third chapter, we construct the Cantor set as a homeomorphic image of the ring of \(p\)-adic integers \(\mathbb{Z}_p\), \(p\) a prime number. First we define \(p\)-adic numbers and whole \(p\)-adic numbers. We define the \(p\)-adic norm and the \(p\)-adic distance on the field \(\mathbb{Q}\) and prove that the field \(\mathbb{Q}\) with the \(p\)-adic the norm is not complete. We complete the field \(\mathbb{Q}\) with the field of equivalence classes of Cauchy sequences, \(\mathbb{Q}_p \). After that, we show the equivalence of the field \(\mathbb{Q}_p \) as a set of equivalence classes of sequences and infinite \(p\)-adic expansions that converge in the \(p\)-adic norm on \(\mathbb{Q} _p\). Finally, we show the existence of a homeomorphism between the ring of \(p\)-adic integers \(\mathbb{Z}_p\) and the Cantor set in the inherited Euclidean topology.

V delu raziščemo kaos v diskretnih dinamičnih sistemih. Najprej definiramo splošne dinamične sisteme in vpeljemo iteracijo preslikave kot klasični zgled diskretnega dinamičnega sistema. Opišemo tri zahteve, ki določajo kaotične diskretne dinamične sisteme: občutljivost na začetne pogoje, gostost periodičnih točk in topološka tranzitivnost. Ogledamo si družino šotorskih preslikav $T_mu$ v odvisnosti od parametra $mu > 0$ in raziščemo njihovo dinamiko. Pri določenih parametrih $mu$ je dinamični sistem, porojen z iteriranjem preslikave $T_mu$, kaotičen. Za parameter $mu = 2$ dokažemo, da sistem zadošča zahtevam kaotičnih dinamičnih sistemov. Za parametre $mu > 2$ vpeljemo simbolni prostor $Sigma_2$ in pokažemo, da je pomik $sigma$ kaotična preslikava. Nato definiramo preslikavo, ki je topološka konjugacija med $T_mu$ in $sigma$ in pokažemo, da ohranja kaotičnost. In this diploma thesis, we explore chaos in discrete dynamical systems. First, we define dynamical systems in general and introduce an iterated map as the prototypical example of a discrete dynamical system. We describe the three conditions that define chaotic discrete dynamical systems: sensitive dependence on initial conditions, dense periodic points and topological transitivity. Then, we examine the dynamics of the one-parameter family of tent maps $T_mu$ where $mu > 0$ and show that the dynamic system arising from iterating $T_mu$ is chaotic for some parameters $mu$. For $mu = 2$, we directly check that the system meets criteria for chaotic systems. For $mu > 2$, we introduce the symbol space $Sigma_2$ and show that the shift map $sigma$ is chaotic. We then define a map that is a topological conjugacy between $T_mu$ and $sigma$ and show that it preserves chaoticity.

Let \(C\) be the middle-third Cantor set. Define \(C*C=\{x*y\colon x,y\in C\}\), where \(*=+,-,\cdot,\div\) (when \(*=\div\), we assume \(y\neq0\)). Steinhaus [17] proved in 1917 that \(C-C=[-1,1]\), \(C+C=[0,2]\). In 2019, Athreya, Reznick and Tyson [1] proved that \(C\div C=\bigcup_{n=-\infty}^{\infty}\left[ 3^{-n}\dfrac{2}{3},3^{-n}\dfrac {3}{2}\right] \cup \{0\}\). In this paper, we give a description of the topological structure and Lebesgue measure of \(C\cdot C\). We indeed obtain corresponding results on the uniform \(\lambda\)-Cantor sets.

For a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi-periodically forced oscillators. These solutions are well-understood in quasi-periodically perturbed nonlinear oscillators either in the presence of large damping or in the non-degenerate cases with small or free damping. In this paper, we consider the existence of response solutions in quasi-periodically perturbed, second order differential equations, including nonlinear oscillators, of the form $$\begin{aligned} \ddot{x}+\lambda x^l=\epsilon f(\omega t,x,\dot{x}),\;\qquad \;x\in \mathbb {R}, \end{aligned}$$ where $$\lambda $$ is a constant, $$01$$ is an integer, $$\omega \in \mathbb {R}^d$$ is a frequency vector, and $$f: \mathbb {T}^d\times \mathbb {R}^2\rightarrow \mathbb {R}^1$$ is real analytic and non-degenerate in x up to a given order $$p\ge 0$$ , i.e., $$[f(\cdot ,0,0)]=[\frac{\partial f(\cdot ,0,0)}{\partial x}]=[\frac{\partial ^2 f(\cdot ,0,0)}{\partial x^2}]=\cdots =[\frac{\partial ^{p-1} f(\cdot ,0,0)}{\partial x^{p-1}}]=0$$ and $$[\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{p}}]\ne 0$$ , where $$[\ \ ]$$ denotes the average value of a continuous function on $$\mathbb {T}^d$$ . In the case that $$\lambda =0$$ and f is independent of $$\dot{x}$$ , the existence of response solutions was first shown by Gentile (Ergod Theory Dyn Syst 27:427–457, 2007) when $$p=1$$ . This result was later generalized by Corsi and Gentile (Commun Math Phys 316:489–529, 2012; Ergod Theory Dyn Syst 35:1079–1140, 2015; Nonlinear Differ Equ Appl 24(1):article 3, 2017) to the case that $$p>1$$ is odd. In the case $$\lambda \ne 0$$ , the existence of response solutions is studied by the authors Si and Yi (Nonlinearity 33(11):6072–6099, 2020) when $$p=0$$ . The present paper is devoted to the study of response solutions of the above quasi-periodically perturbed differential equations for the case $$\lambda \ne 0$$ by allowing $$p>0$$ . Under the conditions that $$0\le p 0$$ when $$l-p$$ is even, we obtain a general result which particularly implies the following: (1) If either l is odd and $$\lambda 0$$ , then as $$\epsilon $$ sufficiently small response solutions exist for each $$\omega $$ satisfying a Brjuno-like non-resonant condition; (2) If either l is odd and $$\lambda >0$$ or l is even and $$[\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{p}}]0$$ sufficiently small and a Cantor set $$\mathcal {E}\in (0,\epsilon _*)$$ with almost full Lebesgue measure such that response solutions exist for each $$\epsilon \in \mathcal {E}$$ and $$\omega $$ satisfying a Diophantine condition. Similar results are also obtained in the case $$\lambda =\pm \epsilon $$ which particularly concern the existence of large amplitude response solutions.

We introduce a connection between Newhouse thickness and patterns through a variant of Schmidt’s game introduced by Broderick, Fishman and Simmons. This yields an explicit, robust and checkable condition that ensures that a Cantor set in the real line contains long arithmetic progressions and, more generally, homothetic copies of large finite sets.

We consider a model of nonlinear wave equations with periodically varying wave speed and periodic external forcing. By imposing non-resonance conditions on the frequency, we establish the existence of response solutions (i.e., periodic solutions with the same frequency as the forcing) for such a model in a Cantor set of asymptotically full measure. The proof relies on a Lyapunov–Schmidt reduction together with the Nash–Moser iteration.

Analogs of Waring–Hilbert problem on Cantor sets are explored. The focus of this paper is on the Cantor ternary set C . It is shown that, for each m ≥ 3 , every real number in the unit interval [ 0 , 1 ] is the sum x 1 m + x 2 m + ⋯ + x n m with each x j in C and some n ≤ 6 m . Furthermore, every real number x in the interval [ 0 , 8 ] can be written as x = x 1 3 + x 2 3 + ⋯ + x 8 3 , the sum of eight cubic powers with each x j in C . Another Cantor set C × C is also considered. More specifically, when C × C is embedded into the complex plane ℂ , the Waring–Hilbert problem on C × C has a positive answer for powers less than or equal to 4.

Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

We introduce and study bi-Lipschitz-invariant dimensions that range between the box and Assouad dimensions. The quasi-Assouad dimensions and $\theta$-spectrum are other special examples of these intermediate dimensions. These dimensions are localized, like Assouad dimensions, but vary in the depth of scale which is considered, thus they provide very refined geometric information. We investigate the relationship between these and the familiar dimensions. We construct a Cantor set with a non-trivial interval of dimensions, the endpoints of this interval being given by the quasi-Assouad and Assouad dimensions of the set. We study continuity-like properties of the dimensions. In contrast with the Assouad-type dimensions, we see that decreasing sets in $\mathbb{R}$ with decreasing gaps need not have dimension $0$ or $1$. Formulas are given for the dimensions of Cantor-like sets and these are used in some of our constructions. We also show that, as is the case for Hausdorff and Assouad dimensions, the Cantor set and the decreasing set have the extreme dimensions among all compact sets in $\mathbb{R}$ whose complementary set consists of open intervals of the same lengths., Comment: Updated reference

S. Marcus used a fat Cantor set to show that there is a quasicontinuous function from the interval [0,1] to R that is not Lebesgue measurable. However, his article is not very well known. We give a...

It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in $$\mathbb {S}^n$$ , for $$n\ge 2$$ . Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if E is a compact, uniformly perfect and uniformly disconnected set in $$\mathbb {S}^n$$ , then it is the Julia set of a hyperbolic UQR map $$f:\mathbb {S}^N \rightarrow \mathbb {S}^N$$ where $$N=n$$ if $$n=2$$ and $$N=n+1$$ otherwise.

We prove that every homogeneous countable dense homogeneous topological space containing a copy of the Cantor set is a Baire space. In particular, every countable dense homogeneous topological vector space is a Baire space. It follows that, for any nondiscrete metrizable space $X$, the function space $C_p(X)$ is not countable dense homogeneous. This answers a question posed recently by R. Hern\'andez-Guti\'errez. We also conclude that, for any infinite dimensional Banach space $E$ (dual Banach space $E^\ast$), the space $E$ equipped with the weak topology ($E^\ast$ with the weak$^\ast$ topology) is not countable dense homogeneous. We generalize some results of Hru\v{s}\'ak, Zamora Avil\'es, and Hern\'andez-Guti\'errez concerning countable dense homogeneous products., Comment: slightly modified and expanded version

Let C be the middle-third Cantor set. In this paper, we show that for every x ∈ [ 0 , 4 ] , there exist x 1 , x 2 , x 3 , x 4 ∈ C such that x = x 1 2 + x 2 2 + x 3 2 + x 4 2 , which was conjectured in Athreya et al. (2019) [1] .

Fractal geometry can be applied in interior architecture by using certain shapes that depend on the principle of self-similarity and repeating the shape to an infinite number of times, and one of the most popular shapes that can be applied is the Cantor group that can be used as a basis in completing the facades and the locations of doors and windows. For snowflakes, which can be used in devising various forms of decoration, also the Serpinsky Triangle, which is considered one of the most important forms of fractal geometry, which can be applied in internal architecture. It is formed by following the molecular fractals of the components of a small molecule. Use it to generate multiple ideas to make shapes for horizontal projections, whether internal or external. يمکن تطبيق هندسة الفراکتال فى العمارة الداخلية عن طريق استخدام أشکال معينه اعتمدت على مبدأ التشابه الذاتى وتکرار الشکل الى عدد لانهائي من المرات , ومن أشهر الأشکال التى يمکن تطبيقها هى مجموعة کانتور التى يمکن استخدامها کأساس فى تصميم الواجهات ومواقع الأبواب والشبابيک , أيضا هناک مايسمى بمنحنى کوخ لرقائق الثلج والذى يمکن استخدامه فى استنباط أشکال متنوعه من الزخارف , أيضا مثلث سيربنسکى وهو يعتبر من أهم أشکال هندسة الفراکتال والتي يمکن تطبيقها فى العمارة الداخلية وهو يتکون عن طريق متابعة الفراکتالات الجزيئية لمکونات جزئ صغير , ومن أهم الأشکال لهندسة الفراکتال أيضا ما يسمى بالتکرارات الهندسية والتي يمکن استخدامها لإنتاج أفکار متعددة لعمل أشکال لمساقط أفقيه سواء داخليه أو خارجية .

In this paper, we consider the following one-dimensional, quasi-periodically forced, linear KdV equations $$\begin{aligned} u_t+(1+ a_{1}(\omega t)) u_{xxx}+ a_{2}(\omega t,x) u_{xx}+ a_{3}(\omega t,x)u_{x} +a_{4}(\omega t,x)u=0 \end{aligned}$$ under the periodic boundary condition $$u(t,x+2\pi )=u(t,x)$$ , where $$\omega $$ ’s are frequency vectors lying in a bounded closed region $$\Pi _*\subset {\mathbb {R}}^b$$ for some $$b>1$$ , $$a_1:{\mathbb {T}}^b\rightarrow {\mathbb {R}}$$ , $$a_i: {\mathbb {T}}^b\times {\mathbb {T}}\rightarrow {\mathbb {R}}$$ , $$i=2,3,4$$ , are real analytic, bounded from the above by a small parameter $$\epsilon _*>0$$ under a suitable norm, and $$a_1,a_3$$ are even, $$a_2,a_4$$ are odd. Under the real analyticity assumption of the coefficients, we re-visit a result of Baldi et al. (Math Ann 359(1–2):471–536, 2014) by showing that there exists a Cantor set $$\Pi _{\epsilon _*}\subset \Pi _*$$ with $$|\Pi _*\setminus \Pi _{\epsilon _*}|=O(\epsilon _*^{\frac{1}{100}})$$ such that for each $$\omega \in \Pi _{\epsilon _*}$$ , the corresponding equation is smoothly reducible to a constant-coefficient one. Our main result removes a condition originally assumed in Baldi et al. (2014) and thus can yield general existence and linear stability results for quasi-periodic solutions of a reversible, quasi-periodically forced, nonlinear KdV equation with much less restrictions on the nonlinearity. The proof of our reducibility result makes use of some special structures of the equations and is based on a refined Kuksin’s estimate for solutions of homological equations with variable coefficients.

We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian which is associated with the standard symplectic structure, where λ = ±1 and n > 2, n ≥ m ≥ 2 are integers. With P satisfying certain non-degenerate conditions, we obtain the following results: (1) For λ = −1 and sufficiently small, responsive tori exist for each ω satisfying a weak non-resonant condition; (2) For λ = 1 and * sufficiently small, there exists a Cantor set with almost full Lebesgue measure such that responsive tori exist for each if ω satisfies a Diophantine condition. Non-existence of responsive tori are also discussed when P fails to satisfy the non-degenerate condition. Our results are directly applicable to the existence problem of quasi-periodic responsive solutions of degenerate harmonic oscillators.

Let $X\in\text{Alex}\,^n(-1)$ be an $n$-dimensional Alexandrov space with curvature $\ge -1$. Let the $r$-scale $(k,\epsilon)$-singular set $\mathcal S^k_{\epsilon,\,r}(X)$ be the collection of $x\in X$ so that $B_r(x)$ is not $\epsilon r$-close to a ball in any splitting space $\mathbb R^{k+1}\times Z$. We show that there exists $C(n,\epsilon)>0$ and $\beta(n,\epsilon)>0$, independent of the volume, so that for any disjoint collection $\big\{B_{r_i}(x_i):x_i\in \mathcal S_{\epsilon,\,\beta r_i}^k(X)\cap B_1, \,r_i\le 1\big\}$, the packing estimate $\sum r_i^k\le C$ holds. Consequently, we obtain the Hausdorff measure estimates $\mathcal H^k(\mathcal S^k_\epsilon(X)\cap B_1)\le C$ and $\mathcal H^n\big(B_r (\mathcal S^k_{\epsilon,\,r}(X))\cap B_1(p)\big)\leq C\,r^{n-k}$. This answers an open question asked by Kapovitch and Lytchak. We also show that the $k$-singular set $\mathcal S^k(X)=\underset{\epsilon>0}\cup\left(\underset{r>0}\cap\mathcal S^k_{\epsilon,\,r}\right)$ is $k$-rectifiable and construct examples to show that such a structure is sharp. For instance, in the $k=1$ case we can build for any closed set $T\subseteq \mathbb S^1$ and $\epsilon>0$ a space $Y\in\text{Alex}^3(0)$ with $\mathcal S^{1}_\epsilon(Y)=\phi(T)$, where $\phi\colon\mathbb S^1\to Y$ is a bi-Lipschitz embedding. Taking $T$ to be a Cantor set it gives rise to an example where the singular set is a $1$-rectifiable, $1$-Cantor set with positive $1$-Hausdorff measure., Comment: 28 pages

Diophantine approximation in dynamical systems concerns the Diophantine properties of the orbits. In classic Diophantine approximation, the powerful mass transference principle established by Beresnevich and Velani provides a general principle to the dimension for a limsup set. In this paper, we aim at finding a general principle for the dimension of the limsup set arising in a general expanding dynamical system. More precisely, let (X, T) be a topological dynamical system where X is a compact metric space and $$T:X\rightarrow X$$ is an expanding continuous transformation. Given $$y_o\in X$$ , we consider the following limsup set $$\mathcal {W}(T,f)$$ , driven by the dynamical system (X, T), $$\begin{aligned} \Big \{x\in X: x\in B(z, e^{-S_n(f+\log |T'|)(z)})\ \text {for some}\ z\in T^{-n}y_o\ {\text {with infinitely many}}\ n\in \mathbb {N}\Big \}, \end{aligned}$$ where $$\log |T'|$$ is a function reflecting the local conformality of the transformation T, f is a non-negative continuous function over X, and $$S_n (f+ \log |T'|) (z)$$ denotes the ergodic sum $$(f+ \log |T'|) (z)+\cdots +(f+ \log |T'|) (T^{n-1} z)$$ . By proposing a dynamical ubiquity property assumed on the system (X, T), we obtain that the dimensions of X and $$\mathcal {W}(T,f)$$ are both related to the Bowen-Manning-McCluskey formulae, namely the solution to the pressure functions $$\begin{aligned} \texttt {P}(-t\log |T'|)=0\ \text {and}\ \texttt {P}(-t(\log |T'|+f))=0, \text {respectively}. \end{aligned}$$ We call this phenomenon a dynamical dimension transference principle, because of its partial analogy with the mass transference principle. This general principle unifies and extends some known results which were considered only separatedly before. These include the b-adic expansions, expanding rational maps over Julia sets, inhomogeneous. Diophantine approximation on the triadic Cantor set and finite conformal iterated function systems.

If f:R3→R3 is a uniformly quasiregular mapping with Julia set J(f), a genus g Cantor set, for g≥1, then for any linearizer L at any repelling periodic point of f, the fast escaping set A(L) consists of a spiders' web structure containing embedded genus g tori on any sufficiently large scale. In other words, A(L) contains a spiders' web of doughnuts. This type of structure is specific to higher dimensions, and cannot happen for the fast escaping set of a transcendental entire function in the plane. We also show that if f:Rn→Rn is a uniformly quasiregular mapping, for n≥2, and J(f) is a Cantor set, then every periodic point is in J(f) and is repelling.

We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a Cantor set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal structure of the limiting Cantor set. The computation of the Extremal Index is connected to the box dimension of the intersection between the Cantor set and its iterates.