scholarly journals The topological entropy of iterated piecewise affine maps is uncomputable

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Pascal Koiran
Keyword(s):  
Cellular Automata ◽  
Real Number ◽  
Topological Entropy ◽  
Linear Functions ◽  
Affine Function ◽  
Piecewise Affine ◽  
Affine Maps ◽  
International Audience

International audience We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is always a computable real number, and conversely whether every non-negative computable real number can be obtained as the topological entropy of a piecewise affine function. It seems that these two questions are also open for cellular automata.

scholarly journals Shifts with decidable language and non-computable entropy

2008 ◽  
Vol Vol. 10 no. 3 (Automata, Logic and Semantics) ◽  
Author(s):  
Peter Hertling ◽  
Christoph Spandl
Keyword(s):  
Real Number ◽  
Topological Entropy ◽  
Polynomial Time ◽  
The Other ◽  
Other Hand ◽  
Enumerable Language ◽  
International Audience ◽  
Full Shift ◽  
The One ◽  
Infinite Sequences

Automata, Logic and Semantics International audience We consider subshifts of the full shift of all binary bi-infinite sequences. On the one hand, the topological entropy of any subshift with computably co-enumerable language is a right-computable real number between 0 and 1. We show that, on the other hand, any right-computable real number between 0 and 1, whether computable or not, is the entropy of some subshift with even polynomial time decidable language. In addition, we show that computability of the entropy of a subshift does not imply any kind of computability of the language of the subshift

Analytical expression of explicit MPC solution via lattice piecewise-affine function

Automatica ◽  
2009 ◽  
Vol 45 (4) ◽  
pp. 910-917 ◽  
Author(s):  
Chengtao Wen ◽  
Xiaoyan Ma ◽  
B. Erik Ydstie
Keyword(s):  
Affine Function ◽  
Analytical Expression ◽  
Piecewise Affine

scholarly journals Prime numbers with a certain extremal type property

2018 ◽  
Vol 17 (1) ◽  
pp. 127-151
Author(s):  
Edward Tutaj
Keyword(s):  
Riemann Hypothesis ◽  
Convex Set ◽  
Counting Function ◽  
Numerical Data ◽  
Prime Numbers ◽  
Affine Function ◽  
Piecewise Affine ◽  
Order Of Magnitude ◽  
Type Property ◽  
Prime Counting Function

Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points $({e_k},\pi ({e_k}))_1^\infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ${{{e_k} + 1} \over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

On the preservation of properties by piecewise affine maps of locally compact groups

2019 ◽  
Vol 12 (3) ◽  
pp. 491-502
Author(s):  
Serina Camungol ◽  
Matthew Morison ◽  
Skylar Nicol ◽  
Ross Stokke
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Piecewise Affine ◽  
Affine Maps

On the geometry and regularity of invariant sets of piecewise-affine automorphisms on the Euclidean space

2019 ◽  
Vol 40 (8) ◽  
pp. 2183-2218
Author(s):  
C. SİNAN GÜNTÜRK ◽  
NGUYEN T. THAO
Keyword(s):  
Euclidean Space ◽  
Rate Of Convergence ◽  
Linear Part ◽  
Invariant Sets ◽  
Measurable Partition ◽  
Ergodic Averages ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Orbit Closures ◽  
Borel Measurable

In this paper, we derive geometric and analytic properties of invariant sets, including orbit closures, of a large class of piecewise-affine maps $T$ on $\mathbb{R}^{d}$. We assume that (i) $T$ consists of finitely many affine maps defined on a Borel measurable partition of $\mathbb{R}^{d}$, (ii) there is a lattice $\mathscr{L}\subset \mathbb{R}^{d}$ that contains all of the mutual differences of the translation vectors of these affine maps, and (iii) all of the affine maps have the same linear part that is an automorphism of $\mathscr{L}$. We prove that finite-volume invariant sets of such piecewise-affine maps always consist of translational tiles relative to this lattice, up to some multiplicity. When the partition is Jordan measurable, we show that closures of bounded orbits of $T$ are invariant and yield Jordan measurable tiles, again up to some multiplicity. In the latter case, we show that compact $T$-invariant sets also consist of Jordan measurable tiles. We then utilize these results to quantify the rate of convergence of ergodic averages for $T$ in the case of bounded single tiles.

Topological entropy of periodic coven cellular automata

2016 ◽  
Vol 36 (2) ◽  
pp. 579-592
Author(s):  
Weibin LIU ◽  
Jihua MA
Keyword(s):  
Cellular Automata ◽  
Topological Entropy

scholarly journals REACHABILITY PROBLEMS IN LOW-DIMENSIONAL ITERATIVE MAPS

2008 ◽  
Vol 19 (04) ◽  
pp. 935-951 ◽  
Author(s):  
OLEKSIY KURGANSKYY ◽  
IGOR POTAPOV ◽  
FERNANDO SANCHO-CAPARRINI
Keyword(s):  
Reachability Problem ◽  
One Dimensional ◽  
Piecewise Affine ◽  
Affine Maps ◽  
Iterative Maps ◽  
Low Dimensional

In this paper we analyze the dynamics of one-dimensional piecewise maps. We show that one-dimensional piecewise affine maps are equivalent to pseudo-billiard or so called “strange billiard” systems. We also show that use of more general classes of functions lead to undecidability of reachability problem for one-dimensional piecewise maps.

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