We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most $\log 2$, and by the Variational Principle, the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only 2 different slopes. For those maps, there is an additional restriction that the topological entropy is larger than $\frac{1}{2}\log2$. We also make the interesting observation that for skew tent maps the sum of reciprocals of derivatives of all iterates of the map at the critical value is zero. It is a generalization and a different interpretation of the Milnor-Thurston formula connecting the topological entropy and the kneading determinant for unimodal maps.

Consider two objects associated to the Iterated Function System (IFS) $\{1+\lambda z,-1+\lambda z\}$: the locus $\mathcal{M}$ of parameters $\lambda\in\mathbb{D}\setminus\{0\}$ for which the corresponding attractor is connected; and the locus $\mathcal{M}_0$ of parameters for which the related attractor contains $0$. The set $\mathcal{M}$ can also be characterized as the locus of parameters for which the attractor of the IFS $\{1+\lambda z, \lambda z, -1+\lambda z\}$ contains $\lambda^{-1}$. Exploiting the asymptotic similarity of $\mathcal{M}$ and $\mathcal{M}_0$ with the respective associated attractors, we give sufficient conditions on $\lambda\in\partial\mathcal{M}$ or $\partial\mathcal{M}_0$ to guarantee it is path accessible from the complement $\mathbb{D}\setminus\mathcal{M}$.

It is well known that Gaussian polynomials (i.e., $q$-binomials) describe the distribution of the $area$ statistic on monotone paths in a rectangular grid. We introduce two new statistics, $corners$ and $cindex$; attach ``ornaments'' to the grid; and re-evaluate these statistics, in order to argue that all scrambled versions of the $cindex$ statistic are equidistributed with $area$. Our main result is a representation of the generating function for the bi-statistic $(cindex,corners)$ as a two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths.

We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.

We study a recursion that generates real sequences depending on a parameter $$x$$ . Given a negative $$x$$ the growth of the sequence is very difficult to estimate due to canceling terms. We reduce the study of the recursion to a problem about a family of integral operators, and prove that for every parameter value except $$-1$$ , the growth of the sequence is factorial. In the combinatorial part of the proof we show that when $$x=-1$$ the resulting recurrence yields the sequence of alternating Catalan numbers, and thus has exponential growth. We expect our methods to be useful in a variety of similar situations.

We present a simple rational map of the complex projective plane whose first and second dynamical degrees coincide, but which does not have any invariant foliation.

Oscillatory regulatory networks have been discovered in many regulatory pathways. Due to their enormous complexity, it is necessary to study their dynamics by means of highly simplified models. These models have received particular value because artificial regulatory networks can be engineered experimentally. In this paper, we study dynamical properties of an artificial regulatory oscillator called repressilator. We have shown that oscillations arise from the existence of an absorbing toruslike region in the phase space of the model. This geometric structure requires monotonic repression at all promoters and the absence of any regulatory connections apart from a cyclic repression loop. We show that oscillations collapse as only weak extra connections are introduced if there is imbalance between the attended concentrations and those sufficient for saturation of the promoters. We found that a pair of diffusively coupled repressilators displays synchronization properties similar to those of relaxation oscillators if the regulatory connections in the cyclic repression loop are strong. Thus, the role of strengthening these connections can be viewed as introducing time scale separation among variables. This may explain controversial synchronization properties reported for repressilators in earlier studies.

The Repressilator is a genetic regulatory network used to model oscillatory behavior of more complex regulatory networks like the circadian clock. We prove that the Repressilator equations undergo a supercritical Hopf bifurcation as the maximal rate of protein synthesis increases, and find a large range of parameters for which there is a cycle.

During a saddle-node bifurcation for real analytic interval maps, a pair of fixed points, attracting and repelling, collide and disappear. From the complex point of view, they do not disappear, but just become complex conjugate. The question is whether those new complex fixed points are attracting or repelling. We prove that this depends on the Schwarzian derivative S S at the bifurcating fixed point. If S S is positive, both fixed points are attracting; if it is negative, they are repelling.

We prove that the sum of greatest common divisors of parts in all compositions of n equals the sum of lengths of all cyclic compositions of n. The proof highlights structural similarities between the set of compositions of n and the set of cyclic compositions of n.

Nekrashevych conjectured that the iterated monodromy groups of quadratic polynomials with preperiodic critical orbit have intermediate growth. We illustrate some of the difficulties that arise in attacking this conjecture and prove subexponential growth for the iterated monodromy group of $z^2+i$. This is the first non-trivial example supporting the conjecture.

We give a recursive formula to count maximal small copies of the Mandelbrot set and its higher degree analogues. This formula is used to compute the asymptotic growth of the number of maximal small copies of period n.

. We present a new identity involving compositions (i.e., ordered partitions of natural numbers). The formula has its origin in complex dynamical systems and appears when counting, in the polynomial family { f_c :z ↦ z^d + c} , periodic critical orbits with equivalent itineraries. We give two different proofs of the identity; one following the original approach in dynamics and another with purely combinatorial methods.

We supplement the definition of principal nest, introduced by M. Lyubich, with a "system of frames" that makes possible the classification of combinatorial types for every level of the nest. As a consequence, we give necessary and sufficient conditions for the admissibility of a type and prove that given a sequence of finite admissible types, there is a quadratic polynomial whose nest realizes the sequence.

Given a critically periodic quadratic map with no secondary renormalizations, we introduce the notion of $Q$-recurrent quadratic polynomials. We show that the pieces of the principal nest of a $Q$-recurrent map $f_c$ converge in shape to the Julia set of $Q$. We use this fact to compute analytic invariants of the nest of $f_c$, to give a complete characterization of complex quadratic Fibonacci maps and to obtain a new auto-similarity result on the Mandelbrot set.