On the measure attractor of a cellular automaton
Petr Kůrka
Conference Publications 2005, 2005(Special): 524-535 doi: 10.3934/proc.2005.2005.524
Given a cellular automaton $F:A^{\ZZ} \to A^{\ZZ}$, we define its small quasi-attractor $\Qq_F$ as the nonempty intersection of all shift-invariant attractors of all $F^q\sigma^p$, where $q>0$ and $p\in\ZZ$. The measure attractor $\Mm_F$ is the closure of the supports of the members of the unique attractor of $F:\MMM_{\sigma}(A^{\ZZ}) \to \MMM_{\sigma}(A^{\ZZ})$ in the space of shift-invariant Borel probability measures.
keywords: invariant subshifts attractors. signals Borel measures
Minimality in iterative systems of Möbius transformations
Petr Kůrka
Conference Publications 2011, 2011(Special): 903-912 doi: 10.3934/proc.2011.2011.903
We study the parameter space of an iterative system consisting of two hyperbolic disc Möbius transformations. We identify several classes of parameters which yield discrete groups whose fundamental polygons have sides at the Euclidean boundary. It follows that these system are not minimal.
keywords: iterative Möbius systems topological dynamics minimal systems
Iterative systems of real Möbius transformations
Petr Kůrka
Discrete & Continuous Dynamical Systems - A 2009, 25(2): 567-574 doi: 10.3934/dcds.2009.25.567
We investigate iterative systems consisting of Möbius transformations on the extended real line. We characterize systems with unique attractor and give some sufficient conditions for minimality.
keywords: iterative Möbius systems. attractors topological dynamics
Dynamically defined recurrence dimension
Petr Kůrka Vincent Penné Sandro Vaienti
Discrete & Continuous Dynamical Systems - A 2002, 8(1): 137-146 doi: 10.3934/dcds.2002.8.137
We modify the idea of a previous article [8] and introduce polynomial and exponential dynamically defined recurrence dimensions, topological invariants which express how the Poincaré recurrence time of a set grows when the diameter of the set shrinks. We introduce also the concept of polynomial entropy which applies in the case that topological entropy is zero and complexity function is polynomial. We compare recurrence dimensions with topological and polynomial entropies, evaluate recurrence dimensions of Sturmian subshifts and show some examples with Toeplitz subshifts.
keywords: polynomial entropy recurrence dimension Toeplitz subshifts. Sturmian subshifts Poincaré recurrence time

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