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We introduce the notion of intrinsic semilattice entropy $\widetilde h$ in the category $\mathcal L_{qm}$ of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories $\mathfrak X$ and functors $F:\mathfrak X\to\mathcal L_{qm}$ we find specific known entropies $\widetilde h_\mathfrak X$ on $\mathfrak X$ as intrinsic functorial entropies, that is, as $\widetilde h_\mathfrak X=\widetilde h\circ F$. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological entropy for locally compact totally disconnected groups and the algebraic entropy for locally compact compactly covered abelian groups.

The notion of the (dual) hull-kernel topology on a collection of prime filters in a residuated lattice is introduced and investigated. It is observed that any collection of prime filters is a $$T_0$$ topological space under the (dual) hull-kernel topology. It is proved that any collection of prime filters is a $$T_1$$ space if and only if it is an antichain, and it is a Hausdorff space if and only if it satisfies some certain conditions. Some characterizations in which maximal filters form a Hausdorff space are given. In the end, we focus on the space of minimal prim filters, and verify that this space is totally disconnected Hausdorff. This paper is closed by description of the compactness of the space of the minimal prime filters using the space of prime $$\alpha $$ -filters.

A λ-graph bisystem L consists of a pair (L−,L+) of two labelled Bratteli diagrams, that presents a two-sided subshift ΛL. We will construct a compact totally disconnected metric space consisting of...

We are interested in a signed graph G˙ which admits a decomposition into a totally disconnected (i.e., without edges) star complement and a signed graph S˙ induced by the star set. In this study we derive certain properties of G˙ ; for example, we prove that the number of (distinct) eigenvalues of S˙ does not exceed the number of those of G˙ . Some particular cases are also considered.

For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

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.

A locally compact contraction group is a pair ( G , α ) , where G is a locally compact group and α : G → G an automorphism such that α n ( x ) → e pointwise as n → ∞ . We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles to construct uncountably many pairwise non-isomorphic totally disconnected, locally compact contraction groups ( G , α ) which are central extensions { 0 } → F p ( ( t ) ) → G → F p ( ( t ) ) → { 0 } of the additive group of the field of formal Laurent series over F p = Z / p Z by itself. By contrast, there are only countably many locally compact contraction groups (up to isomorphism) which are torsion groups and abelian, as follows from a classification of the abelian locally compact contraction groups.

A locally compact group G is said to be approximable by totally disconnected subgroups if there is a sequence of closed totally disconnected subgroups $$(\Gamma _n)_{n\in {\mathbb N}}$$ of G such that for any non-empty open set O of G, there exists an integer k such that $$O\cap \Gamma _n \ne \varnothing $$ , for every $$n\ge k$$ . In this paper, we prove that every pro-torus is approximable by profinite subgroups and we provide a characterization of the approximation of compact groups. Furthermore, we show that every compactly generated locally compact abelian group is approximable by totally disconnected subgroups. As an application, a study of the Chabauty space of closed totally disconnected subgroups is given.

In this paper, we prove that, under Bandt's finite type condition, a self-similar set is totally disconnected if and only if the \(E\)-connected families on every level are uniformly finite.

Let $\mathcal {G}$ be a second countable, Hausdorff topological group. If $\mathcal {G}$ is locally compact, totally disconnected and T is an expansive automorphism then it is shown that the dynamical system $(\mathcal {G}, T)$ is topologically conjugate to the product of a symbolic full-shift on a finite number of symbols, a totally wandering, countable-state Markov shift and a permutation of a countable coset space of $\mathcal {G}$ that fixes the defining subgroup. In particular if the automorphism is transitive then $\mathcal {G}$ is compact and $(\mathcal {G}, T)$ is topologically conjugate to a full-shift on a finite number of symbols.

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups S on discrete abelian groups A by endomorphisms; they extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case S = N . We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.

Let be a metric space and be a continuous map. The notion of the -average shadowing property ( ASP ) for a continuous map on –space is introduced and the relation between the ASP and average shadowing property(ASP)is investigated. We show that if has ASP, then has ASP for every . We prove that if a map be pseudo-equivariant with dense set of periodic points and has the ASP, then is weakly mixing. We also show that if is a expansive pseudo-equivariant homeomorphism that has the ASP and is topologically mixing, then has a -specification. We obtained that the identity map on has the ASP if and only if the orbit space of is totally disconnected. Finally, we show that if is a pseudo-equivariant map, and the trajectory map is a covering map, then has the ASP if and only if the induced map has ASP.

In this paper we study the hyperspace of all nonempty closed totally disconnected subsets of a space, equipped with the Vietoris topology. We show results of compactness, connectedness and local connectedness for this hyperspace. We also include a study of path connectedness, particularly we prove that for a smooth dendroid this hyperspace is pathwise connected, and we present a general result which implies that for an Euclidean space this hyperspace has uncountably many arc components.

For a locally compact metrizable group G, we consider the action of $$\mathrm{Aut}(G)$$ on $$\mathrm{Sub}_G$$ , the space of all closed subgroups of G endowed with the Chabauty topology. We study the structure of groups G admitting automorphisms T which act expansively on $$\mathrm{Sub}_G$$ . We show that such a group G is necessarily totally disconnected, T is expansive and that the contraction groups of T and $$T^{-1}$$ are closed and their product is open in G; moreover, if G is compact, then G is finite. We also obtain the structure of the contraction group of such T. For the class of groups G which are finite direct products of $${\mathbb {Q}}_p$$ for distinct primes p, we show that $$T\in \mathrm{Aut}(G)$$ acts expansively on $$\mathrm{Sub}_G$$ if and only if T is expansive. However, any higher dimensional p-adic vector space $${\mathbb {Q}}_p^n$$ , ( $$n\ge 2$$ ), does not admit any automorphism which acts expansively on $$\mathrm{Sub}_G$$ .

In this work we introduced new types of -light functions namely -light functions, also we introduced -disconnected, and -totally disconnected spaces. Some examples, facts, and theorems have been given to support our work.

By the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. In the case of two independent random variables we give a complete description of compact totally disconnected Abelian groups X, where an analogue of this theorem is valid. We also prove that even a weak analogue of the Heyde theorem fails on compact connected Abelian groups X. Coefficients of considered linear forms are topological automorphisms of X. The proofs are based on the study of solutions of a functional equation on the character group of the group X in the class of Fourier transforms of probability distributions.

For a locally compact group 𝐺, we write S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) for the space of closed subgroups of 𝐺 endowed with the Chabauty topology. For any positive integer 𝑛, we associate to 𝐺 the function δ G , n \delta_{G,n} from G n G^{n} to S ⁢ U ⁢ B ⁢ ( G ) {\mathcal{SUB}}(G) defined by δ G , n ⁢ ( g 1 , … , g n ) = gp ¯ ⁢ ( g 1 , … , g n ) , \delta_{G,n}(g_{1},\ldots,g_{n})=\overline{\mathrm{gp}}(g_{1},\ldots,g_{n}), where gp ¯ ⁢ ( g 1 , … , g n ) \overline{\mathrm{gp}}(g_{1},\ldots,g_{n}) denotes the closed subgroup topologically generated by g 1 , … , g n g_{1},\ldots,g_{n} . It would be interesting to know for which groups 𝐺 the function δ G , n \delta_{G,n} is continuous for every 𝑛. Let [ HW ] [\mathtt{HW}] be the class of such groups. Some interesting properties of the class [ HW ] [\mathtt{HW}] are established. In particular, we prove that [ HW ] [\mathtt{HW}] is properly included in the class of totally disconnected locally compact groups. The class of totally disconnected locally compact locally pronilpotent groups is included in [ HW ] [\mathtt{HW}] . Also, we give an example of a solvable totally disconnected locally compact group not contained in [ HW ] [\mathtt{HW}] .

UDC 517.537.38 We prove that each totally disconnected closed subset E of a domain G in the complex plane is removable for analytic functions f ( z ) defined in G ∖ E and such that for any point z 0 ∈ E the real or imaginary part of f ( z ) vanishes at z 0 .

We briefly survey the Hilbert--Smith Conjecture, and we include a proof of it in dimension two (where it is originally due to Montgomery--Zippin)., Comment: 7 pages, 2 figures

A locally compact abelian group is called periodic if it is totally disconnected and is a directed union of its compact subgroups. Various aspects of abelian periodic groups are considered such as – decomposing them into local products of their Sylow p- subgroups, – providing new descriptions of periodic abelian torsion groups, and of – periodic abelian divisible groups and their torsion-free and their torsion components, – reviewing splitting theorems, notably for finite rank pure subgroups of (almost) finite exponent p-groups, – providing a definition of a general p-rank for all locally compact abelian p-groups.

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable $C^{\ast }$-algebra is the stationary inductive limit of a $C^{\ast }$-stable Fell algebra that has a compact spectrum and trivial Dixmier–Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to, in principle, compute the $K$-theory of the stable $C^{\ast }$-algebra. A specific one-dimensional Smale space (the $aab/ab$-solenoid) is considered as an illustrative running example throughout.

Let ϕ : G → H {\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair ( G , ϕ - 1 ⁢ ( L ) ) {(G,\phi^{-1}(L))} , where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.

Fractal theory is the study of irregularity which occurs in natural objects. It also enables us to see patterns in the highly complex and unpredictable structures resulting from many natural phenomena, using self-similarity property. The most common mathematical method to generate self-similar fractals is using an iterated function system (IFS). This paper discusses separation properties of finite products of hyperbolic IFSs. Characterizations for totally disconnected and overlapping product IFSs are obtained. A method to generate an open set which satisfies the open set condition for a totally disconnected IFS is given. Some necessary and sufficient conditions for a product IFS to be just touching are discussed. Also, Type 1 homogenous IFSs are introduced and its separation properties in terms of the separation properties of coordinate projections are explained towards the end.

Let $A$ be an expanding $d\times d$ matrix with integer entries and ${\mathcal D}\subset {\mathbb Z}^d$ be a finite digit set. Then the pair $(A, {\mathcal D})$ defines a unique integral self-affine set $K=A^{-1}(K+{\mathcal D})$. In this paper, by replacing the Euclidean norm with a pseudo-norm $w$ in terms of $A$, we construct a hyperbolic graph on $(A, {\mathcal D})$ and show that $K$ can be identified with the hyperbolic boundary. Moreover, if $(A, {\mathcal D})$ safisfies the open set condition, we also prove that two totally disconnected integral self-affine sets are Lipschitz equivalent if an only if they have the same $w$-Hausdorff dimension, that is, their digit sets have equal cardinality. We extends some well-known results in the self-similar sets to the self-affine sets.

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{*}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given., Comment: 17 pages

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

Graphs considered here are simple, finite and undirected. A graph is denoted by [Formula: see text] and its vertex set by [Formula: see text] and edge set by [Formula: see text]. Kulli and Janakiram introduced the concept of the strong non-split domination number of a graph [The strong non-split domination number of a graph, Int. J. Manag. Syst. 19 (2003) 2]. The concept of fair domination was introduced by Yaro Caro et al. in [Fair domination in graphs, Discrete Math. 312 (2012) 2905–2914]. This paper is an outcome of the combination of these two concepts.

It is known that exactness for a discrete group is equivalent to C*-exactness, i.e., the exactness of its reduced C*-algebra. The problem of whether this equivalence holds for general locally compact groups has recently been reduced by Cave and Zacharias to the case of totally disconnected unimodular groups. We prove that the equivalence does hold for a class of groups that includes all locally compact groups with reduced C*-algebra admitting a tracial state. As a consequence, we present original proofs that totally disconnected locally compact (tdlc) invariant-neighbourhood (IN) groups and a class of groups introduced by Suzuki satisfy the equivalence problem, without using inner amenability.

A locally compact group G is called compactly ruled if it is a directed union of compact open subgroups. We denote by $${\mathcal {SUB}}\left( G\right) $$ the space of closed subgroups of G equipped with the Chabauty topology. In this paper, we show that the subspace $${\mathcal {SUB}}_{\scriptscriptstyle \mathrm {co}}\left( G\right) $$ of $${\mathcal {SUB}}\left( G\right) $$ consisting of compact open subgroups is dense in $${\mathcal {SUB}}\left( G\right) $$ if and only if G is totally disconnected compactly ruled.

We shall prove that if X is a Hausdorff paracompactum with at least one nontrivial quasicomponent, then none of its Cech systems can be commutative and none can support the structure of an approximate inverse system.

We discuss a Plancherel formula for countable groups, which provides a canonical decomposition of the regular representation of such a group $\Gamma$ into a direct integral of factor representations. Our main result gives a precise description of this decomposition in terms of the Plancherel formula of the FC-center $\Gamma_{\rm fc}$ of $\Gamma$ (that is, the normal sugbroup of $\Gamma$ consisting of elements with a finite conjugacy class); this description involves the action of an appropriate totally disconnected compact group of automorphisms of $\Gamma_{\rm fc}$. As an application, we determine the Plancherel formula for linear groups. In an appendix, we use the Plancherel formula to provide a unified proof for Thoma's and Kaniuth's theorems which respectively characterize countable groups which are of type I and those whose regular representation is of type II., Comment: 26 pages

Let R be a commutative ring with identity and M a unitary R-module. The primary-like spectrum SpecL(M) is the collection of all primary-like submodules Q of M, the recent generalization of primary ideals, such that M/Q is a primeful R-module. In this article, we topologies SpecL(M) with the patch-like topology, and show that when, SpecL(M) with the patch-like topology is a quasi-compact, Hausdorff, totally disconnected space.

The aim of this work is to give a pointfree description of the Cantor set. It can be shown that the Cantor set is homeomorphic to the $p$-adic integers $\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p}: |x|_p\leq 1\}$ for every prime number $p$. To give a pointfree description of the Cantor set, we specify the frame of $\mathbb{Z}_{p}$ by generators and relations. We use the fact that the open balls centered at integers generate the open subsets of $\mathbb{Z}_{p}$ and thus we think of them as the basic generators; on this poset we impose some relations and then the resulting quotient is the frame of the Cantor set $\mathcal{L}(\mathbb{Z}_{p})$. We prove that $\mathcal{L}(\mathbb{Z}_{p})$ is a spatial frame whose space of points is homeomorphic to $\mathbb{Z}_{p}$. In particular, we show with pointfree arguments that $\mathcal{L}(\mathbb{Z}_{p})$ is $0$-dimensional, (completely) regular, compact, and metrizable (it admits a countably generated uniformity). Finally, we give a point-free counterpart of the Hausdorff-Alexandroff Theorem which states that \emph{every compact metric space is a continuous image of the Cantor space} (see, e.g. \cite{Alexandroff} and \cite{Hausdorff}). We prove the point-free analog: if $L$ is a compact metrizable frame, then there is an injective frame homomorphism from $L$ into $\mathcal{L}(\mathbb{Z}_{2})$., 1 figure

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio&rsquo, s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.

In this paper we investigate the block numerical range and the existences of estimable decompositions of bounded linear operators on a separable Hilbert space. By using spectral measure, we show that there exists an estimable decomposition for the spectrum of every bounded normal operator. Furthermore, the corresponding result also holds for hyponormal operators with totally disconnected spectra. Finally, we obtain that for spectral operator, there exists an estimable decomposition, under quasi-nilpotent equivalence.

Wieler showed that every irreducible Smale space with totally disconnected local stable sets is an inverse limit system, called a Wieler solenoid. We study self-similar inverse semigroups defined by s-resolving factor maps of Wieler solenoids. We show that the groupoids of germs and the tight groupoids of these inverse semigroups are equivalent to the unstable groupoids of Wieler solenoids. We also show that the C &lowast, algebras of the groupoids of germs have a unique tracial state.

We prove that the $L^2$-Betti numbers of a rigid $C^*$-tensor category vanish in the presence of an almost-normal subcategory with vanishing $L^2$-Betti numbers, generalising a result of Bader, Furman and Sauer. We apply this criterion to show that the categories constructed from totally disconnected groups by Arano and Vaes have vanishing $L^2$-Betti numbers. Given an almost-normal inclusion of discrete groups $\Lambda, Comment: 44 pages, 1 figure. v2: Minor corrections. Added a remark and an example

We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1., Comment: Abstract slightly reformulated

In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings R, in particular for R = Z and R = Q . We show these properties satisfy many analogous results to the case of discrete groups, and we provide analogues of the famous Bieri's and Brown's criteria for finiteness properties and deduce that both FP n -properties and F n -properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness properties.

Let X_t be a totally disconnected subset of the real line R for each t in R. We construct a partition {Y_t | t in R} of R into nowhere dense Lebesgue null sets Y_t such that for every t in R there exists an increasing homeomorphism from X_t onto Y_t. In particular, the real line can be partitioned into 2^{aleph_0} Cantor sets and also into 2^{aleph_0} mutually non-homeomorphic compact subspaces. Furthermore we prove that for every cardinal number k with 2 \leq k \leq 2^{aleph_0} the real line (as well as the Baire space R\Q) can be partitioned into exactly k homeomorphic Bernstein sets and also into exactly k mutually non-homeomorphic Bernstein sets. We also investigate partitions of R into Marczewski sets, including the possibility that they are Luzin sets or Sierpinski sets.

An omitted value of a transcendental meromorphic function $f$ is called a Baker omitted value, in short \textit{bov} if there is a disk $D$ centered at the bov such that each component of the boundary of $f^{-1}(D)$ is bounded. Assuming that the bov is in the Fatou set of $f$, this article investigates the dynamics of the function. Firstly, the connectivity of all the Fatou components are determined. If $U$ is the Fatou component containing the bov then it is proved that a Fatou component $U'$ is infinitely connected if and only if it lands on $U$, i.e. $f^{k}(U') \subset U$ for some $k \geq 1$. Every other Fatou component is either simply connected or lands on a Herman ring. Further, assuming that the number of critical points in the Fatou set whose forward orbits do not intersect $U$ is finite, we have shown that the connectivity of each Fatou component belongs to a finite set. This set is independent of the Fatou components. It is proved that the Fatou component containing the bov is completely invariant whenever it is forward invariant. Further, if the invariant Fatou component is an attracting domain and compactly contains all the critical values of the function then the Julia set is totally disconnected. Baker domains are shown to be non-existent whenever the bov is in the Fatou set. It is also proved that, if there is a $2$-periodic Baker domain (these are not ruled out when the bov is in the Julia set), or a $2$-periodic attracting or parabolic domain containing the bov then the function has no Herman ring. Some examples exhibiting different possibilities for the Fatou set are discussed. This includes the first example of a meromorphic function with an omitted value which has two infinitely connected Fatou components., Comment: 21 pages, 7 figures

International audience; We introduce and explore a natural rank for totally disconnected locally compact groups called the bounded conjugacy rank. This rank is shown to be a lattice invariant for lattices in sigma compact totally disconnected locally compact groups; that is to say, for a given sigma compact totally disconnected locally compact group, some lattice has bounded conjugacy rank n if and only if every lattice has bounded conjugacy rank n. Several examples are then presented.

In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions that are not differentiable or integrable on totally disconnected fractal sets such as middle-mu Cantor sets. Analogues of the Lyapunov functions and their features are given for asymptotic behaviors of fractal differential equations. The stability of fractal differentials in the sense of Lyapunov is defined. For the suggested fractal differential equations, sufficient conditions for the stability and uniform boundedness and convergence of the solutions are presented and proved. We present examples and graphs for more details of the results.

Let ${\cal M}_{t,v,r}(n,m)$, $2\leq m, Comment: arXiv admin note: text overlap with arXiv:2005.07451

In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $\{ T_t(z)=i t\tan z\}_{0< t\leq ��}$. Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as $t$ increases we find single instances of "period quadrupling", "period splitting" and standard "period doubling"; there follows a general pattern of "period merging" where two attracting cycles of period $2^n$ "merge" into one attracting cycle of period $2^{n+1}$, and "cycle doubling" where an attracting cycle of period $2^{n+1}$ "becomes" two attracting cycles of the same period. We use renormalization to prove the existence of these bifurcation parameters. The uniqueness of the cycle doubling and cycle merging parameters is quite subtle and requires a new approach. To prove the cycle doubling and merging parameters are, indeed, unique, we apply the concept of "holomorphic motions" to our context. In addition, we prove that there is an "infinitely renormalizable" tangent map $T_{t_\infty}$. It has no attracting or parabolic cycles. Instead, it has a strange attractor contained in the real and imaginary axes which is forward invariant and minimal under $T^2_{t_\infty}$. The intersection of this strange attractor with the real line consists of two binary Cantor sets and the intersection with the imaginary line is totally disconnected, perfect and unbounded., 52 pages, 15 figures

It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group $G$ the same information about the number of ends of $G$ in the sense of H. Abels can be provided by $\mathrm{dH}^1(G,\mathrm{Bi}(G))$, where $\mathrm{Bi}(G)$ is the rational discrete standard bimodule of $G$, and $\mathrm{dH}^\bullet(G,\_)$ denotes rational discrete cohomology as introduced in [6]. As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B)., more detailed version - some corrections have been made -title has been changed