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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.

Let $M$ be a compact manifold of dimension at least 2. If $M$ admits a minimal homeomorphism then $M$ admits a minimal noninvertible map.

In this article we show that R.H. Bing’s pseudo-circle admits a minimal non-invertible map. This resolves a conjecture raised by Bruin, Kolyada and Snoha in the negative. The main tool is a variant of the Denjoy–Rees technique, further developed by Béguin–Crovisier–Le Roux, combined with detailed study of the structure of the pseudo-circle. This is the first example of a planar 1-dimensional space that admits both minimal homeomorphisms and minimal noninvertible maps.

We provide several new examples in dynamics on the $2$-sphere, with the emphasis on better understanding the induced boundary dynamics of invariant domains in parametrized families. First, motivated by a topological version of the Poincare-Bendixson Theorem obtained recently by Koropecki and Passeggi, we show the existence of homeomorphisms of $\mathbb{S}^2$ with Lakes of Wada rotational attractors, with an arbitrarily large number of complementary domains, and with or without fixed points, that are arbitrarily close to the identity. This answers a question of Le Roux. Second, from reduced Arnold's family we construct a parametrised family of Birkhoff-like cofrontier attractors, where at least for uncountably many choices of the parameters, two distinct irrational prime ends rotation numbers are induced from the two complementary domains. This example complements the resolution of Walker's Conjecture by Koropecki, Le Calvez and Nassiri from 2015. Third, answering a question of Boyland, we show that there exists a non-transitive Birkhoff-like attracting cofrontier which is obtained from a BBM embedding of inverse limit of circles, such that the interior prime ends rotation number belongs to the interior of the rotation interval of the cofrontier dynamics. There exists another BBM embedding of the same attractor so that the two induced prime ends rotation numbers are exactly the two endpoints of the rotation interval.

We show that every (invertible, or noninvertible) minimal Cantor system embeds in $\mathbb{R}$ with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.

In 2005 Kawamura and Rambla, independently, constructed a metric counterexample to Wood's Conjecture from 1982. We exhibit a new nonmetric counterexample of a space L ˆ , such that C 0 ( L ˆ , C ) is almost transitive, and show that it is distinct from a nonmetric space whose existence follows from the work of Greim and Rajagopalan in 1997. Up to our knowledge, this is only the third known counterexample to Wood's Conjecture. We also show that, contrary to what was expected, if a one-point compactification of a space X is R.H. Bing's pseudo-circle then C 0 ( X , C ) is not almost transitive, for a generic choice of points. Finally, we point out close relation of these results on Wood's conjecture to a work of Irwin and Solecki on projective Fraisse limits and projective homogeneity of the pseudo-arc and, addressing their conjecture, we show that the pseudo-circle is not approximately projectively homogeneous.

We show that the Cartesian product of the arc and a solenoid has the fupcon property, therefore answering a question raised by Illanes. This combined with Illanes' result implies that the product of a Knaster continuum and a solenoid has the fupcon property, therefore answering a question raised by Bellamy and Łysko in the affirmative. Finally, we show that a product of two Smith's nonmetric pseudo-arcs has the fupcon property.

We prove that the Sierpinski curve admits a homeomorphism with strong mixing properties. We also prove that the constructed example does not have Bowen's specification property.

Motivated by a recent result of Ciesielski and Jasinski we study periodic point free Cantor systems that are conjugate to systems with vanishing derivative everywhere, and more generally locally radially shrinking maps. Our study uncovers a whole spectrum of dynamical behaviors attainable for such systems, providing new counterexamples to the Conjecture of Edrei from 1952, first disproved by Williams in 1954.

We prove that there exists a topologically mixing homeomorphism which is completely scrambled. We also prove that, for any integer$n\geq 1$, there is a continuum of topological dimension$n$supporting a transitive completely scrambled homeomorphism and an$n$-dimensional compactum supporting a weakly mixing completely scrambled homeomorphism. This solves a 15-year-old open problem.

We study nonmetric analogues of Vietoris solenoids. Let $$\Lambda $$ be an ordered continuum, and let $$\vec {p}=\langle p_1,p_2,\ldots \rangle $$ be a sequence of positive integers. We define a natural inverse limit space $$S(\Lambda ,\vec {p})$$ , where the first factor space is the nonmetric “circle” obtained by identifying the endpoints of $$\Lambda $$ , and the nth factor space, $$n>1$$ , consists of $$p_1p_2 \ldots p_{n-1}$$ copies of $$\Lambda $$ laid end to end in a circle. We prove that for every cardinal $$\kappa \ge 1$$ , there is an ordered continuum $$\Lambda $$ such that $$S(\Lambda ,\vec {p})$$ is $$\frac{1}{\kappa }$$ -homogeneous; for $$\kappa >1$$ , $$\Lambda $$ is built from copies of the long line. Our example with $$\kappa =2$$ provides a nonmetric answer to a question of Neumann-Lara, Pellicer-Covarrubias and Puga from 2005, and with $$\kappa =1$$ provides an example of a nonmetric homogeneous circle-like indecomposable continuum. We also show that for each uncountable cardinal $$\kappa $$ and for each fixed $$\vec {p}$$ , there are $$2^\kappa $$ -many $$\frac{1}{\kappa }$$ -homogeneous solenoids of the form $$S(\Lambda ,\vec {p})$$ as $$\Lambda $$ varies over ordered continua of weight $$\kappa $$ . Finally, we show that for every ordered continuum $$\Lambda $$ the shape of $$S(\Lambda ,\vec {p})$$ depends only on the equivalence class of $$\vec {p}$$ for a relation similar to one used to classify the additive subgroups of $$\mathbb {Q}$$ . Consequently, for each fixed $$\Lambda $$ , as $$\vec {p}$$ varies, there are exactly $$\mathfrak {c}$$ -many different shapes, where $$\mathfrak {c}=2^{\aleph _0}$$ , (and there are also exactly that many homeomorphism types) represented by $$S(\Lambda ,\vec {p})$$ .

We show that for any map f on an arc-like continuum X, the induced map f ˆ on the hyperspace of subcontinua C ( X ) fixes a point in any f ˆ -invariant subcontinuum of C ( X ) . This extends a result of Robatian [21] , who proved it for the arc. However, as we show, the result does not extend to tree-like continua. We conclude with a list of related problems. Our proof builds on Hamilton's proof of the fixed point property of arc-like continua [12] .

In 1985 M. Smith constructed a nonmetric pseudo-arc; i.e. a Hausdorff homogeneous, hereditary equivalent and hereditary indecomposable continuum. Taking advantage of a decomposition theorem of W. Lewis, he obtained it as a long inverse limit of metric pseudo-arcs with monotone bonding maps. Extending his approach, and the results of Lewis on lifting homeomorphisms, we construct a nonmetric pseudo-circle, and new examples of homogeneous one-dimensional continua; e.g. a circle and solenoids of nonmetric pseudo-arcs. Among many corollaries we also obtain an analogue of another theorem of Lewis from 1984: any interval map is semi-conjugate to a homeomorphism of the nonmetric pseudo-arc.

Given a collection of pairwise co-prime integers $% m_{1},\ldots ,m_{r}$, greater than $1$, we consider the product $\Sigma =\Sigma _{m_{1}}\times \cdots \times \Sigma _{m_{r}}$, where each $\Sigma _{m_{i}}$ is the $m_{i}$-adic solenoid. Answering a question of D. P. Bellamy and J. M. Łysko, in this paper we prove that if $M$ is a subcontinuum of $\Sigma $ such that the projections of $M$ on each $\Sigma _{m_{i}}$ are onto, then for each open subset $U$ in $\Sigma $ with $M\subset U$, there exists an open connected subset $V$ of $\Sigma $ such that $M\subset V\subset U$; i.e. any such $M$ is ample in the sense of Prajs and Whittington [10]. This contrasts with the property of Cartesian squares of fixed solenoids $\Sigma _{m_{i}}\times \Sigma _{m_{i}}$, whose diagonals are never ample [1].

The result of Boyce and Huneke gives rise to a 1-dimensional continuum, which is the intersection of a descending family of disks, that admits two commuting homeomorphisms without a common fixed point., to appear in Proceedings of the American Mathematical Society

We exhibit a Li-Yorke chaotic interval map F such that the inverse limit XF = lim ←−{F, [0, 1]} does not contain an indecomposable subcontinuum. Our result contrasts with the known property of interval maps: if φ has positive entropy then Xφ contains an indecomposable subcontinuum. Each subcontinuum of XF is homeomorphic to one of the following: an arc, or XF , or a topological ray limiting to XF . From a result of Barge and Martin it follows that XF is a chaotic attractor of a planar homeomorphism. In addition, F can be modified to give a cofrontier that is a chaotic attractor of a planar homeomorphism but contains no indecomposable subcontinuum. Finally, F can be modified, without removing or introducing new periods, to give a chaotic zero entropy interval map, such that the corresponding inverse limit contains the pseudoarc.

A continuum X is called solenoidal if it is circle-like and nonplanar. X is 1 n -homogeneous if the action of its homeomorphism group Homeo ( X ) on X has exactly n orbits; i.e. there are exactly n types of points in X . Recently Jimenez-Hernandez, Minc and Pellicer-Covarrubias [6] constructed a family of 1 n -homogeneous solenoidal continua, for every n > 2 . Modifying the spaces obtained by them, as well as an earlier construction of the author for n = 2 , for every n > 2 we construct two different uncountable families of arcless 1 n -homogeneous solenoidal continua Σ n and Σ n ′ . We also show that there is an uncountable family of countably nonhomogeneous solenoidal continua Σ ∞ ; i.e. each Y ∈ Σ ∞ has (infinitely) countably many types of points. For every Y ∈ ⋃ n ∈ N Σ n ∪ Σ ∞ any orbit of Homeo ( Y ) is uncountable. With respect to the degree of homogeneity, in the realm of solenoidal continua containing pseudoarcs, our examples complete the gap between homogeneous solenoids of pseudoarcs and uncountably nonhomogeneous pseudosolenoids. A number of questions related to the study is raised.

We construct a 2-torus homeomorphism \(h\) homotopic to the identity with an attracting R. H. Bing’s pseudocircle \(C\) such that the rotation set of \(h|C\) is not a unique vector. The only known examples of such an attractor were Birkhoff’s attractors, arising for dissipative maps with a twist. Our construction relies heavily on the Barge–Martin method for constructing attractors as inverse limits of graphs.

Fox theory of overlays is used to characterize finite-sheeted covering spaces of pseudosolenoids (i.e. hereditarily indecomposable circle-like nonchainable continua). Let S be a P -adic pseudosolenoid, for a sequence of prime numbers P = ( p 1 , p 2 , … , p n , … ) . S admits a d -fold cover onto itself if and only if d is relatively prime to all but finitely many p i . Moreover, if C is any connected finite-sheeted covering space of S then C is homeomorphic to S . These results parallel known results for solenoids and extend results of Bellamy and Heath for the pseudocircle. In addition, it is shown that most self-maps of pseudosolenoids are finite-sheeted covering maps. Namely, if E ( S ) is the space of all surjective self-maps of S with the sup metric, then the subset C ( S ) of local self-homeomorphisms of S is a dense G δ in E ( S ) . This result, based on a theorem of Kawamura, extends the near-homeomorphism property of the pseudoarc proved independently by Lewis and Smith, and relates to a question raised by Lewis in 1984, as to which other nondegenerate continua have the near-homeomorphism property.

Using an earlier fixed point result of the author and Nielsen-type arguments patterned on related results for torus self-maps by Alseda et al. we describe minimal sets of periods for self-maps of cofrontiers that are inverse limits of circles. Special attention is given to the pseudocircle, ever present in dynamical systems. Our result extends a property of circle maps, proved by Efremova, and independently by Block et al. in the late 1970s. We explain why our natural generalization is rather unexpected and shows potential of Nielsen theory for new applications.

We exhibit an indecomposable 1 2 -homogeneous circle-like continuum, answering a question of Neumann-Lara, Pellicer-Covarrubias and Puga-Espinosa from 2005. Our solenoidal construction is an extension of Jim Rogersʼ proof of homogeneity of solenoids of pseudoarcs and leads in turn to an uncountable family of mutually nonequivalent examples. In addition we prove that for an indecomposable 1 2 -homogeneous continuum Y the two orbits of Homeo ( Y ) are uncountable.

G ianfrancesco Malfatti (Figure 1) was a brilliant Italian mathematician born in 1731 in a small village in the Italian Alps, Ala, near Trento. He first studied at a Jesuit school in Verona, then at the University of Bologna. Malfatti was one of the founders of the Department of Mathematics of the University of Ferrara. He died in Ferrara in 1807. As a very active intellectual in the Age of Enlightenment, he devoted himself to the promotion of many new ideas and wrote many papers in different fields of mathematics including algebra, calculus, geometry, and probability theory. He played an important role in the creation of the Nuova Enciclopedia Italiana (1779), in the spirit of the French Encyclopedie edited by Diderot and d’Alembert. His mathematical papers were collected by the Italian Mathematical Society in the volume [7]. His historical figure has been discussed in a series of papers in [1]. This paper was inspired by a conference in 2007 commemorating the 200th anniversary of Malfatti’s death, organized by the municipality of Ala and the mathematics departments of Ferrara and Trento. Malfatti appears in the mathematical literature of the last two centuries mostly in connection with a problem he raised and discussed in a paper in 1803 (Figure 2): how to pack three non-overlapping circles of maximum total area in a given triangle? Malfatti assumed that the solution consisted of three mutually tangent circles, each also tangent to two edges of the triangle (now called Malfatti’s arrangement) and in his paper he constructed such arrangements (for a historical overview see [3]). In 1994 Zalgaller and Los [11] disproved Malfatti’s original assumption and showed that the greedy arrangement is always the best one. The detailed story of this 200-year-old problem is worth telling because it has many paradigms typical of research in mathematics, including the way one formulates a problem, how one interprets it or solves it, and what one should consider trivial and what one should not. In the following section we give the history of the problem. The section after that contains a new non-analytic solution for the problem of maximizing the total area of two disjoint circles contained in a given triangle. In the last section we generalize the two-circle problem for certain regions other than triangles. Our non-analytic approach shows that in various situations the greedy arrangements are the best ones.

Two results concerning orientation-reversing homeomorphisms of the plane are proved. Let h : R 2 → R 2 h:\mathbb {R}^2\rightarrow \mathbb {R}^2 be an orientation-reversing planar homeomorphism with a continuum X X invariant (i.e. h ( X ) = X h(X)=X ). First, suppose there are at least n n bounded components of R 2 ∖ X \mathbb {R}^2\setminus X that are invariant under h h . Then there are at least n + 1 n+1 components of the fixed point set of h h in X X . This provides an affirmative answer to a question posed by K. Kuperberg. Second, suppose there is a k k -periodic orbit in X X with k > 2 k>2 . Then there is a 2-periodic orbit in X X , or there is a 2-periodic component of R 2 ∖ X \mathbb {R}^2\setminus X . The second result is based on a recent result of M. Bonino concerning linked periodic orbits of orientation-reversing homeomorphisms of the 2-sphere S 2 \mathbb {S}^2 . These results generalize to orientation-reversing homeomorphisms of S 2 \mathbb {S}^2 .

A set (or a collection of sets) contained in the Euclidean space R m is symmetric if it is invariant under the antipodal map. Given a symmetric unicoherent polyhedron X (like an n-dimensional cube or a sphere) and an odd real function f defined on vertices of a certain symmetric triangulation of X, we algorithmically construct a connected symmetric separator of X by choosing a subcollection of the triangulation. Each element of the subcollection contains the vertices v and u such that f ( v ) f ( u ) ⩽ 0 .

Within this contribution we establish a theoretical background necessary for studying inverse limits of fuzzy dynamical systems induced by crisp (non-fuzzy) ones. First, we elaborate topological properties of the space of fuzzy sets, such as connectedness and compactness. Second, we focus our attention on dynamical conditions sufficient for the existence of indecomposable continua in the inverse limit space.

A new combinatorial result intertwined with the Brouwer fixed point theorem for the n-cube is given. This result can be used for any map (f 1, ..., fn): [0, 1] n → [0, 1] n to approximate the components of the set {(x 1, . . . , x n ) ∈ [0, 1] n : f i (x 1, . . . , x n ) = x i } that separate the n-cube between the i th opposite faces. Equivalently, for maps g : [0, 1] n → ℝ such that g(x)g(y) ≤ 0 for any x ∈ {0} × [0, 1] n-1and y ∈ {1} × [0, 1] n-1, one can use the algorithm to approximate the components of g -1(0) that separate [0, 1] n between {0} × [0, 1] n-1and {1} × [0, 1] n-1. The methods are based on an earlier result of P. Minc and the present authors and relate to results of several other authors such as Jayawant and Wong, Kulpa and Turzański, and Gale. Mathematics Subject Classification (2000): Primary 54H25; 54-04; Secondary 55M20; 54F55.

We prove that if $$f:G\rightarrow G$$ is a map on a topological graph G such that the inverse limit $$\varprojlim (G,f)$$ is hereditarily indecomposable, and entropy of f is positive, then there exists an entropy set with infinite topological entropy. When G is the circle and the degree of f is positive then the entropy is always infinite and the rotation set of f is nondegenerate. This shows that the Anosov-Katok type constructions of the pseudo-circle as a minimal set in volume-preserving smooth dynamical systems, or in complex dynamics, obtained previously by Handel, Herman and Cheritat cannot be modeled on inverse limits. This also extends a previous result of Mouron who proved that if $$G=[0,1]$$ , then $$h(f)\in \{0,\infty \}$$ , and combined with a result of Ito shows that certain dynamical systems on compact finite-dimensional Riemannian manifolds must either have zero entropy on their invariant sets or be non-differentiable.

We show that for every positive integer k there exists an interval map $f:I\to I$ such that (1) f is Li-Yorke chaotic, (2) the inverse limit space $I_{f}=\lim_{\leftarrow}\{f,I\}$ does not contain an indecomposable subcontinuum, (3) f is $C^{k}$ -smooth, and (4) f is not $C^{k+1}$ -smooth. We also show that there exists a $C^{\infty}$ -smooth f that satisfies (1) and (2). This answers a recent question of Oprocha and the first author from (Proc. Am. Math. Soc. 143(8):3659-3670, 2015), where the result was proved for $k=0$ . Our study builds on the work of Misiurewicz and Smital of a family of zero entropy weakly unimodal maps. With the help of a result of Bennett, as well as Blokh’s spectral decomposition theorem, we are also able to show that each $I_{f}$ contains, for every integer i, a subcontinuum $C_{i}$ with the following two properties: (i) $C_{i}$ is $2^{i}$ -periodic under the shift homeomorphism, and (ii) $C_{i}$ is a compactification of a topological ray. Finally, we prove that the chaotic attractors we construct are topologically distinct from the one presented by P Oprocha and the first author.