Given $\varepsilon> 0$ and a bounded Lipschitz domain $\Omega$ in $\mathbb R^M\times \mathbb R^N$ let $\Omega_\varepsilon:=\{(x,\varepsilon y)\mid (x,y)\in \Omega\}$ be the $\varepsilon$-squeezed domain. Consider the reaction-diffusion equation $$ u_t = \Delta u + f(u) \leqno(\widetilde E_\varepsilon) $$ on $\Omega_\varepsilon$ with Neumann boundary condition. Here $f$ is an appropriate nonlinearity such that $(\widetilde E_\varepsilon)$ generates a (local) semiflow $\widetilde\pi_ \varepsilon$ on $H^1(\Omega_\varepsilon)$. It was proved by Prizzi and Rybakowski (J. Differential Equations, to appear), generalizing some previous results of Hale and Raugel, that there are a closed subspace $H^1_s(\Omega)$ of $H^1(\Omega)$, a closed subspace $L^2_s(\Omega)$ of $L^2(\Omega)$ and a sectorial operator $A_0$ on $L^2_s(\Omega)$ such that the semiflow $\pi_0$ defined on $H^1_s(\Omega)$ by the abstract equation $$\dot u+A_0u=\widehat f(u)$$ is the limit of the semiflows $\widetilde\pi_\varepsilon$ as $\varepsilon\to 0^+$.

In this paper we prove a singular Conley index continuation principle stating that every isolated invariant set $K_0$ of $\pi_0$ can be continued to a nearby family $\widetilde K_\varepsilon$ of isolated invariant sets of $\widetilde \pi_\varepsilon$ with the same Conley index. We present various applications of this result to problems like connection lifting or resonance bifurcation.

We present a proof of the non-solvability in radicals of a general algebraic equation of degree greater than four. This proof relies on the non-solvability of the monodromy group of a general algebraic function.

The domain identification problems for the elliptic hemivariational inequalities are studied. These problems are formulated as the optimal control problems with admissible domains as controls. The existence of optimal shapes is obtained by the direct method of calculus of variations for a l.s.c. cost functional.

In [A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree, Ann. Sci. Math. Québec 22 (1998), 131–148] we introduced a concept of orientation and topological degree for nonlinear Fredholm maps between real Banach manifolds. In this paper we study properties of this notion of orientation and we compare it with related results due to Elworthy-Tromba and Fitzpatrick-Pejsachowicz-Rabier.

We prove that there is a coincidence index for the inclusion $F(x)\in\Phi(x)$ when $\Phi$ is convex-valued and satisfies certain compactness assumptions on countable sets. For $F$ we assume only that it provides a coincidence index for single-valued finite-dimensional maps (e.g. $F$ is a Vietoris map). For the special case $F={\rm id}$, the obtained fixed point index is defined if $\Phi$ is countably condensing; the assumptions in this case are even weaker than in [M. Väth, Fixed point theorems and fixed point index for countably condensing maps, Topol. Methods Nonlinear Anal. 13 (1999), 341–363].

In this paper known results on the existence of pure Nash equilibria in games with nonatomic measure space of players are generalized and also a simple proof is offered. The relaxed assumptions include metrizability of the space of actions, measurability of payoff functions and available strategy correspondences.

The theory of quasifactors of minimal dynamical systems is surveyed and several new examples based on ideas of H. Furstenberg are introduced. In particular the open question whether a minimal quasifactor of a minimal proximal system is necessarily proximal is answered in the negative.

To every homeomorphism $T$ of a Cantor set $X$ one can associate the full group $[T]$ formed by all homeomorphisms $\gamma$ such that $\gamma(x)=T^{n(x)}(x)$, $ x\in X$. The topological full group $[[T]]$ consists of all homeomorphisms whose associated orbit cocycle $n(x)$ is continuous. The uniform and weak topologies, $\tau_u$ and $\tau_w$, as well as their intersection $\tau_{uw}$ are studied on ${\rm Homeo}(X)$. It is proved that $[[T]]$ is dense in $[T]$ with respect to $\tau_u$. A Cantor minimal system $(X,T)$ is called saturated if any two clopen sets of "the same measure" are $[[T]]$-equivalent. We describe the class of saturated Cantor minimal systems. In particular, $(X,T)$ is saturated if and only if the closure of $[[T]]$ in $\tau_{uw}$ is $[T]$ and if and only if every infinitesimal function is a $T$-coboundary. These results are based on a description of homeomorphisms from $[[T]]$ related to a given sequence of Kakutani-Rokhlin partitions. It is shown that the offered method works for some symbolic Cantor minimal systems. The tool of Kakutani-Rokhlin partitions is used to characterize $[[T]]$-equivalent clopen sets and the subgroup $[[T]]_x \subset [[T]]$ formed by homeomorphisms preserving the forward orbit of $x$.