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Register a new userForcing among patterns with no block structure
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Define the following order among all natural numbers except for 2 and 1: [ 4gg 6gg 3gg dots gg 4ngg 4n+2gg 2n+1gg 4n+4ggdots ] Let $f$ be a continuous interval map. We show that if $mgg s$ and $f$ has a cycle with no division (no block structure) of period $m$ then $f$ has also a cycle with no division (no block structure) of period $s$. We describe possible sets of periods of cycles of $f$ with no division and no block structure.
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A tournament H is quasirandom-forcing if the following holds for every sequence (G_n) of tournaments of growing orders: if the density of H in G_n converges to the expected density of H in a random tournament, then (G_n) is quasirandom. Every transitive tournament with at least 4 vertices is quasirandom-forcing, and Coregliano et al. [Electron. J. Combin. 26 (2019), P1.44] showed that there is also a non-transitive 5-vertex tournament with the property. We show that no additional tournament has this property. This extends the result of Bucic et al. [arXiv:1910.09936] that the non-transitive tournaments with seven or more vertices do not have this property.
A cyclic permutation $pi:{1, dots, N}to {1, dots, N}$ has a emph{block structure} if there is a partition of ${1, dots, N}$ into $k otin{1,N}$ segments (emph{blocks}) permuted by $pi$; call $k$ the emph{period} of this block structure. Let $p_1<dots <p_s$ be periods of all possible block structures on $pi$. Call the finite string $(p_1/1,$ $p_2/p_1,$ $dots,$ $p_s/p_{s-1}, N/p_s)$ the {it renormalization tower of $pi$}. The same terminology can be used for emph{patterns}, i.e., for families of cycles of interval maps inducing the same (up to a flip) cyclic permutation. A renormalization tower $mathcal M$ emph{forces} a renormalization tower $mathcal N$ if every continuous interval map with a cycle of pattern with renormalization tower $mathcal M$ must have a cycle of pattern with renormalization tower $mathcal N$. We completely characterize the forcing relation among renormalization towers. Take the following order among natural numbers: $ 4gg 6gg 3gg dots gg 4ngg 4n+2gg 2n+1ggdots gg 2gg 1 $ understood in the strict sense. We show that the forcing relation among renormalization towers is given by the lexicographic extension of this order. Moreover, for any tail $T$ of this order there exists an interval map for which the set of renormalization towers of its cycles equals $T$.
We consider a class of parametrically forced Hamiltonian systems with one-and-a-half degrees of freedom and study the stability of the dynamics when the frequency of the forcing is relatively high or low. We show that, provided the frequency of the forcing is sufficiently high, KAM theorem may be applied even when the forcing amplitude is far away from the perturbation regime. A similar result is obtained for sufficiently low frequency forcing, but in that case we need the amplitude of the forcing to be not too large; however we are still able to consider amplitudes of the forcing which are outside of the perturbation regime. Our results are illustrated by means of numerical simulations for the system of a forced cubic oscillator. In addition, we find numerically that the dynamics are stable even when the forcing amplitude is very large (beyond the range of validity of the analytical results), provided the frequency of the forcing is taken correspondingly low.
We introduce the study of forcing sets in mathematical origami. The origami material folds flat along straight line segments called creases, each of which is assigned a folding direction of mountain or valley. A subset $F$ of creases is forcing if the global folding mountain/valley assignment can be deduced from its restriction to $F$. In this paper we focus on one particular class of foldable patterns called Miura-ori, which divide the plane into congruent parallelograms using horizontal lines and zig-zag vertical lines. We develop efficient algorithms for constructing a minimum forcing set of a Miura-ori map, and for deciding whether a given set of creases is forcing or not. We also provide tight bounds on the size of a forcing set, establishing that the standard mountain-valley assignment for the Miura-ori is the one that requires the most creases in its forcing sets. Additionally, given a partial mountain/valley assignment to a subset of creases of a Miura-ori map, we determine whether the assignment domain can be extended to a locally flat-foldable pattern on all the creases. At the heart of our results is a novel correspondence between flat-foldable Miura-ori maps and $3$-colorings of grid graphs.
We present a comprehensive mechanism for the emergence of rotational horseshoes and strange attractors in a class of two-parameter families of periodically-perturbed differential equations defining a flow on a three-dimensional manifold. When both parameters are zero, its flow exhibits an attracting heteroclinic network associated to two periodic solutions. After slightly increasing both parameters, while keeping a two-dimensional connection unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the periodic solutions do not intersect. We prove a wide range of dynamical behaviour, ranging from an attracting quasi-periodic torus to rotational horseshoes and Henon-like strange attractors. We illustrate our results with an explicit example.
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