We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations...

Let (I,T) be the inverse limit space of a post-critically finite tent map. Conditions are given under which these inverse limit spaces are pairwise nonhomeomorphic. This extends results of Barge & Diamond [2].

We survey recent papers on the problem of backward dynamics in economics, providing along the way a glimpse at the economics perspective, a discussion of the economic models and mathematical tools involved, and a list of applicable literature in both mathematics and economics.

We investigate generalizations of Ingram's Conjecture involving maps on trees. We show that for a class of tentlike maps on the k-star with periodic critical orbit, different maps in the class have distinct inverse limit spaces. We do this by showing that such maps satisfy the conclusion of the Pseudo-isotopy Conjecture, i.e., if h is a homeomorphism of the inverse limit space, then there is an integer N such that h and σ̂^N switch composants in the same way, where σ̂ is the standard shift map of...

We derive several properties of unimodal maps having only periodic points whose period is a power of 2. We then consider inverse limits on intervals using a single strongly unimodal bonding map having periodic points whose only periods are all the powers of 2. One such mapping is the logistic map, $f_{\lambda }\left(x\right)$ = 4λx(1-x) on [f(λ),λ], at the Feigenbaum limit, λ ≈ 0.89249. It is known that this map produces an hereditarily decomposable inverse limit with only three topologically different subcontinua. Other...

This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval exchange transformation. Rauzy graphs language can express many important combinatorial and some dynamical properties. In this case combinatorial properties are considered as being generated by a substitutional system, and dynamical properties are considered...

Gliomas are highly invasive brain tumors that exhibit high and spatially heterogeneous cell proliferation and motility rates. The interplay of proliferation and migration dynamics plays an important role in the invasion of these malignant tumors. We analyze the regulation of proliferation and migration processes with a lattice-gas cellular automaton (LGCA). We study and characterize the influence of the migration/proliferation dichotomy (also known...

Let $H^d$ be the set of all rational maps of degree d ≥ 2 on the Riemann sphere, expanding on their Julia set. We prove that if $f\in H^d$ and all, or all but one, critical points (or values) are in the basin of immediate attraction to an attracting fixed point then there exists a polynomial in the component H(f) of $H^d$ containing f. If all critical points are in the basin of immediate attraction to an attracting fixed point or a parabolic fixed point then f restricted to the Julia set is conjugate to the shift...

We study the jumps of topological entropy for $C^r$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f\in C^r\left([0,1]\right)$ with $h_{top}\left(f\right)>\left(log⁺\right||f^{\text{'}}{\left|\right|}_{\infty })/r$. To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^r$ interval maps.

We give necessary and sufficient conditions for a language to be the language of finite words that occur infinitely many times in an infinite word.

We give necessary and sufficient conditions for a language to be the language of finite words that occur infinitely many times in an infinite word.

We consider a hierarchy of notions of largeness for subsets of ℤ (such as thick sets, syndetic sets, IP-sets, etc., as well as some new classes) and study them in conjunction with recurrence in topological dynamics and ergodic theory. We use topological dynamics and topological algebra in βℤ to establish connections between various notions of largeness and apply those results to the study of the sets $R_{A,B}^{\epsilon }=n\in \mathbb{Z}:\mu \left(A\cap TⁿB\right)>\mu \left(A\right)\mu \left(B\right)-\epsilon$ of times of “fat intersection”. Among other things we show that the sets $R_{A,B}^{\epsilon }$ allow one to distinguish...

We show that a natural quotient of the projective Fraïssé limit of a family that consists of finite rooted trees is the Lelek fan. Using this construction, we study properties of the Lelek fan and of its homeomorphism group. We show that the Lelek fan is projectively universal and projectively ultrahomogeneous in the class of smooth fans. We further show that the homeomorphism group of the Lelek fan is totally disconnected, generated by every neighbourhood of the identity, has a dense conjugacy...