The Devil’s Staircase Dimensions and Measure-theoretical Entropy of Maps with Horizontal Gap
نویسندگان
چکیده
This work elucidates the measure-theoretical entropy and dimensions of a unimodal map with a horizontal gap. The measure-theoretical entropy and dimensions of the Ft (which is defined later )are shown to form a devil’s staircase structure with respect to the gap size t. Pesin’s formula for gap maps is also considered .
منابع مشابه
On the entropy devil’s staircase in a family of gap-tent maps
To analyze the trade-off between channel capacity and noise-resistance in designing dynamical systems to pursue the idea of communications with chaos, we perform a measure theoretic analysis the topological entropy function of a “gap-tent map” whose invariant set is an unstable chaotic saddle of the tent map. Our model system, the “gap-tent map” is a family of tent maps with a symmetric gap, wh...
متن کاملA formula for the fractal dimension d ∼ 0 . 87 of the Cantorian set underlying the Devil ’ s staircase associated with the Circle Map
The Cantor set complementary to the Devil’s Staircase associated with the Circle Map has a fractal dimension d ∼ 0.87, universal for a wide range of maps, such results being of a numerical character. In this paper we deduce a formula for such dimensional value, the corresponding theoretical reasoning permits conjecturing on the nature of its universality. The Devil’s Staircase associated with t...
متن کاملDevil's Staircase of Gap Maps
This study demonstrates the devil's staircase structure of topological entropy functions for one-dimensional symmetric unimodal maps with a gap inside. The results are obtained by using kneading theory and helpful in investigating the communication of chaos. Work partially supported by the NSC under Grant No. 89-2115-M-008-029 and the National Center for Theoretical Sciences Mathematics Divisio...
متن کاملChip-Firing And A Devil’s Staircase
The devil’s staircase – a continuous function on the unit interval [0,1] which is not constant, yet is locally constant on an open dense set – is the sort of exotic creature a combinatorialist might never expect to encounter in “real life.” We show how a devil’s staircase arises from the combinatorial problem of parallel chip-firing on the complete graph. This staircase helps explain a previous...
متن کاملExact Solution of Frenkel-Kontorova Models with a Complete Devil’s Staircase in Higher Dimensions
We solve exactly a class of Frenkel-Kontorova models with piecewise parabolic potential, which has d sub-wells in a period. With careful analysis, we show that the phase diagram of the minimum enthalpy configurations exhibits the structure of a complete d-dimensional devil’s staircase. The winding number of a minimum enthalpy configuration is locked to rational values, while the fraction of ato...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2003