Symbolic dynamics for the Rössler folded towel map
Symbolic extensions in intermediate smoothness on surfaces
We prove that maps with on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].
Symmetric CNS trinomials.
Système dynamique à spectre discret et pavage périodique associé à une substitution
On donne une condition combinatoire effective suffisante pour que le sytème dynamique associé à une substitution de type Pisot ait un spectre purement discret. Dans le cas unimodulaire, cette condition est nécessaire dès que la substitution n'a qu'un cobord trivial ; elle est vérifiée si et seulement si le fractal de Rauzy associé à la substitution engendre un pavage auto-similaire et périodique. On en déduit des conditions de connexité des fractals de Rauzy.
Szemerédi theorem implies Furnsternberg theorem
Tail fields generated by symbol counts in measure-preserving systems
A finite-state stationary process is called (one- or two-sided) super-K if its (one- or two-sided) super-tail field-generated by keeping track of (initial or central) symbol counts as well as of arbitrarily remote names-is trivial. We prove that for every process (α,T) which has a direct Bernoulli factor there is a generating partition β whose one-sided super-tail equals the usual one-sided tail of β. Consequently, every K-process with a direct Bernoulli factor has a one-sided super-K generator....
The conjugacy map over a Sturmian word.
The classification of tiling space flows.
The entropy conjecture for diffeomorphisms away from tangencies
We prove that every diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive.
The forcing relation for horseshoe braid types.
The Language of Caring: Quantitating Medical Practice Patterns using Symbolic Dynamics
Real-world medical decisions rarely involve binary Ðsole condition present or absent- patterns of patient pathophysiology. Similarly, provider interventions are rarely unitary in nature: the clinician often undertakes multiple interventions simultaneously. Conventional approaches towards complex physiologic derangements and their associated management focus on the frequencies of joint appearances, treating the individual derangements of physiology...
The nonexistence of universal metric flows
We consider dynamical systems of the form where is a compact metric space and is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract -limit sets, answering a question by Will Brian.
The number of binary rotation words
We consider binary rotation words generated by partitions of the unit circle to two intervals and give a precise formula for the number of such words of length n. We also give the precise asymptotics for it, which happens to be Θ(n4). The result continues the line initiated by the formula for the number of all Sturmian words obtained by Lipatov [Problemy Kibernet. 39 (1982) 67–84], then independently by Mignosi [Theoret. Comput. Sci. 82 (1991) 71–84], and others.
The Tribonacci substitution.
The universal minimal system for the group of homeomorphisms of the Cantor set
Each topological group G admits a unique universal minimal dynamical system (M(G),G). For a locally compact noncompact group this is a nonmetrizable system with a rich structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one-point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. We show that for the topological group G = Homeo(E) of self-homeomorphisms...
The Williams conjecture is false for irreducible subshifts.
Theoretical and computational bounds for m-cycles of the 3n+1-problem
Three complexity functions
For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes.
Three complexity functions
For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes.
Three complexity functions
For an extensive range of infinite words, and the associated symbolic dynamical systems, we compute, together with the usual language complexity function counting the finite words, the minimal and maximal complexity functions we get by replacing finite words by finite patterns, or words with holes.