Topological Entropy Bounds for Hyperbolic Plateaus of the Henon Map

Abstract

Combining two existing rigorous computational methods, for verifying hyperbolicity (due to Arai) and for computing topological entropy bounds (due to Day et al.), we prove lower bounds on topological entropy for 43 hyperbolic plateaus of the Hénon map. We also examine the 16 area-preserving plateaus studied by Arai and compare our results with related work. Along the way, we augment the algorithms of Day et al. with routines to optimize the algorithmic parameters and simplify the resulting semi-conjugate subshift.

Efficient Automation of Index Pairs in Computational Conley Index Theory

“…Second, in contrast with previous results for Hénon-like systems (e.g. [DFT08,Fro10]), the standard map is a volume-preserving map which exhibits both hyperbolic and elliptic behavior, which make the task of automating the construction of index pairs considerably more difficult. Finally, we generalize the methods of [DFT08] to work on manifolds with nontrivial topologies and not just R n , and our application to the standard map exemplifies this.…”

“…For example, for parameter-depending systems, automation allows for the rigorous exploration of a system at many parameters. Such applications of [DFT08] have already been carried out in [Fro10]. Besides exploration at different values of parameters, automated methods may permit us to construct index pairs where it would be otherwise impossible to do so by hand.…”

“…In [DFT08, §3.2] step (3) was completely solved, in the sense that given any compact set (index pair) K which satisfies the properties required by (2), a completely automated routine was given to prove a semi-conjugacy. Steps (1) and (2), the creation of the index pair, were also fully automated in [DFT08, §3.1] (and [Fro10]), largely due to the properties of "well-behaved" maps like the Hénon map: it has a low-dimensional attractor and all its periodic orbits exhibit hyperbolic-like behavior from a computational perspective. In other words, in such cases there is a dominating invariant set, namely the attractor whose periodic orbits all seem hyperbolic and are easily localizable.…”

Sofic Shifts via Conley Index Theory: Computing Lower Bounds on Recurrent Dynamics for Maps

“…[Fro14, Algorithm 1]). Amalgamations can significantly reduce the number of symbols needed to express a subshift, but are especially useful when trying to compare the produced subshift to another, as in [Fro14,Theorem 5.2].…”

Optimal state amalgamation is NP-hard

“…If the number of regions |A| is too large, one may wish to simplify its representation while preserving this geometric meaning, as well as the Markov property. For example, automated proof systems routinely produce shifts on hundreds of symbols [10,12]. A natural way to simplify these shifts is by coarsening the partition, i.e.…”