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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].

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.