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

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.

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 $$S(\Lambda ,\vec {p})$$ .

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.

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.

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 .