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Critical curves of a piecewise linear map
John A G Roberts1, Asaki Saito2, Franco Vivaldi3
1School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia.
Chaos (Woodbury, N.Y.)
|August 3, 2021
Summary
This study explores parameters in planar maps where orbits return to the boundary. These parameters form algebraic curves linked to symbolic dynamics, revealing geometric properties.
Area of Science:
- Dynamical Systems
- Geometric Analysis
- Symbolic Dynamics
Background:
- Investigating parameter spaces of dynamical systems is crucial for understanding system behavior.
- Planar maps with piecewise linear dynamics present complex behaviors, particularly concerning boundary recurrences.
Purpose of the Study:
- To characterize the parameter space of planar maps linear on half-planes.
- To identify and analyze the set of parameters for which all orbits recur to the boundary.
- To explore the algebraic and geometric properties of these parameter curves.
Main Methods:
- Analysis of planar maps defined piecewise linearly on right and left half-planes.
- Utilizing symbolic dynamics to define itineraries connecting boundary points.
- Investigating the algebraic curves formed by recurrence parameters.
Main Results:
- The set of parameters ensuring boundary recurrence forms algebraic curves.
- These curves are intrinsically linked to the symbolic dynamics of boundary point itineraries.
- The study elucidates the algebraic and geometric characteristics of these recurrence-defining curves.
Conclusions:
- The recurrence behavior in this family of planar maps is precisely governed by algebraic curves.
- Symbolic dynamics provides a powerful framework for understanding the geometry of these parameter sets.
- This research offers insights into the interplay between dynamics, algebra, and geometry in planar maps.
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