IN THIS WORK WE STUDY THE DIFFERENTIABILITY PROPERTIES OF THE CONJUGATION IN THE BARREIRA-VALLS VERSION OF THE HARTMAN-GROBMAN THEOREM FOR NON-AUTONOMOUS AND DELAYED SYSTEMS. INDEED, WE SHOW THAT THE CONJUGACY IN THE BARREIRA-VALLS THEOREM IS A DIFFEOMORPHISM IF WE IMPOSE SOME EXTRA HYPOTHESIS RELATED WITH THE DECAY OF THE PERTURBATION.

IN THIS NOTE WE PROVE THAT A HOMEOMORPHISM IS COUNTABLY-EXPANSIVE IF AND ONLY IF IT IS MEASURE-EXPANSIVE. THIS RESULT IS APPLIED FOR SHOWING THAT THE C1-INTERIOR OF THE SETS OF EXPANSIVE, MEASURE-EXPANSIVE AND CONTINUUM-WISE EXPANSIVE C1-DIFFEOMORPHISMS COINCIDE.

WE CONSIDER THE HAMILTONIAN FUNCTION DEFINED BY THE CUBIC POLYNOMIAL H = 1/2(Y(1)(2) + Y(2)(2)) + V(X(1), X(2)) WHERE THE POTENTIAL V(X) = DELTA V-2(X(1), X(2)) + V-3(X(1), X(2)), WITH V-2(X(1), X(2)) = 1/2(X(1)(2) + X(2)(2)) AND V-3(X(1), X(2)) = 1/3X(1)(3) + F X(1)X(2)(2) + GX(2)(3), WITH F AND G ARE REAL PARAMETERS SUCH THAT F NOT EQUAL 0 AND DELTA IS 0 OR 1. OUR OBJECTIVE IS TO STUDY THE NUMBER AND BIFURCATIONS OF THE EQUILIBRIA AND ITS TYPE OF STABILITY. MOREOVER, WE OBTAIN THE EXISTENCE OF PERIODIC SOLUTIONS CLOSE TO SOME EQUILIBRIUM POINTS AND AN ISOLATED SYMMETRIC PERIODIC SOLUTION DISTANT OF THE EQUILIBRIA FOR SOME CONVENIENT REGION OF THE PARAMETERS. WE POINT OUT THE ROLE OF THE PARAMETERS AND THE DIFFERENCE BETWEEN THE HOMOGENEOUS POTENTIAL CASE (DELTA = 0) AND THE GENERAL CASE (DELTA = 1).

WE STUDY THE DYNAMICS OF A FAMILY OF PERTURBED THREE-DEGREE-OF-FREEDOM HAMILTONIAN SYSTEMS WHICH ARE IN 1:1:1 RESONANCE. THE PERTURBATION CONSISTS OF AXIALLY SYMMETRIC CUBIC AND QUARTIC ARBITRARY POLYNOMIALS. OUR ANALYSIS IS PERFORMED BY NORMALISATION, REDUCTION AND KAM TECHNIQUES. FIRSTLY, THE SYSTEM IS REDUCED BY THE AXIAL SYMMETRY, AND THEN, PERIODIC SOLUTIONS AND KAM 3-TORI OF THE FULL SYSTEM ARE DETERMINED FROM THE RELATIVE EQUILIBRIA. NEXT, THE OSCILLATOR SYMMETRY IS EXTENDED BY NORMALISATION UP TO TERMS OF DEGREE 4 IN RECTANGULAR COORDINATES; AFTER TRUNCATION OF HIGHER ORDERS AND REDUCTION TO THE ORBIT SPACE, SOME RELATIVE EQUILIBRIA ARE ESTABLISHED AND PERIODIC SOLUTIONS AND KAM 3-TORI OF THE ORIGINAL SYSTEM ARE OBTAINED. AS A THIRD STEP, THE REDUCTION IN THE TWO SYMMETRIES LEADS TO A ONE-DEGREE-OF-FREEDOM SYSTEM THAT IS COMPLETELY ANALYSED IN THE TWICE REDUCED SPACE. ALL THE RELATIVE EQUILIBRIA TOGETHER WITH THE STABILITY AND PARAMETRIC BIFURCATIONS ARE DETERMINED. MOREOVER, THE INVARIANT 2-TORI (RELATED TO THE CRITICAL POINTS OF THE TWICE REDUCED SPACE), SOME PERIODIC SOLUTIONS AND THE KAM 3-TORI, ALL CORRESPONDING TO THE FULL SYSTEM, ARE ESTABLISHED. ADDITIONALLY, THE BIFURCATIONS OF EQUILIBRIA OCCURRING IN THE TWICE REDUCED SPACE ARE RECONSTRUCTED AS QUASI-PERIODIC BIFURCATIONS INVOLVING 2-TORI AND PERIODIC SOLUTIONS OF THE FULL SYSTEM.

WE EXTEND THE CONCEPT OF EXPANSIVE MEASURE [2] FROM HOMEOMORPHISM TO FLOWS. WE PROVE FOR CONTINUOUS FLOWS ON COMPACT SPACES THAT EVERY EXPANSIVE MEASURE HAS NO SINGULARITIES IN THE SUPPORT, IS APERIODIC, IS EXPANSIVE WITH RESPECT TO TIME-T MAPS (BUT NOT CONVERSELY), REMAINS EXPANSIVE UNDER TOPOLOGICAL EQUIVALENCE, VANISHES ALONG THE ORBITS AND IS NATURAL UNDER SUSPENSIONS. WE APPLY THESE PROPERTIES TO PROVE THAT THERE ARE NO EXPANSIVE FLOWS (IN THE SENSE OF [26]) OF ANY CLOSED SURFACE.

WE STUDY THE NOTION OF EXPANSIVITY FOR BOTH HOMEOMORPHISMS AND MEASURES ON 2-METRIC SPACES [8]. AT FIRST GLANCE WE SHOW THAT THERE ARE INFINITE COMPACT CONTINUOUS 2-METRIC SPACES EXHIBITING EXPANSIVE HOMEOMORPHISMS IN THE 2-METRIC SENSE (ROUGHLY SPEAKING 2-METRIC EXPANSIVE HOMEOMORPHISMS). NEXT WE PROVE THE ABSENCE OF EXPANSIVE MEASURES IN THE 2-METRIC SENSE (OR 2-METRIC EXPANSIVE MEASURES) FOR HOMEOMORPHISMS OF SK (K = 1,2) EQUIPPED WITH THE STANDARD TRIANGLE-AREA A INDUCED BY RK+1. WE THEN CONCLUDE THAT THERE ARE NO 2-METRIC EXPANSIVE HOMEOMORPHISMS OF (SK,A) FOR K = 1,2. FINALLY, IT IS PROVED THAT THE SET OF THE SET OF HETEROCLINIC POINTS FOR 2-METRIC EXPANSIVE HOMEOMORPHISMS ON COMPACT CONTINUOUS 2-METRIC SPACES IS COUNTABLE. THIS EXTENDS A WELL-KNOWN RESULT BY REDDY [19].

WE PROVE THAT EVERY FINITE-EXPANSIVE HOMEOMORPHISM WITH THE SHADOWING PROPERTY HAS A KIND OF STABILITY. THIS STABILITY WILL BE GOOD ENOUGH TO IMPLY BOTH THE SHADOWING PROPERTY AND THE DENSENESS OF PERIODIC POINTS IN THE CHAIN RECURRENT SET. NEXT WE ANALYZE THE N-SHADOWING PROPERTY WHICH IS REALLY STRONGER THAN THE MULTISHADOWING PROPERTY IN CHERKASHIN AND KRYZHEVICH (TOPOL METHODS NONLINEAR ANAL 50(1): 125?150, 2017). WE SHOW THAT AN EQUICONTINUOUS HOMEOMORPHISM HAS THE N-SHADOWING PROPERTY FOR SOME POSITIVE INTEGER N IF AND ONLY IF IT HAS THE SHADOWING PROPERTY.

""""EN LAS MONOGRAFÍAS SE PUBLICAN TEMAS ACTUALES DE INVESTIGACIÓN ESCRITOS DE UNA MANERA EXPOSITIVA POR EXPERTOS EN EL ÁREA. ESTÁN DESTINADOS A SERVIR COMO INICIO EN TEMAS NUEVOS DE INVESTIGACIÓN EN MATEMÁTICA. EL IMCA Y EL FONDO EDITORIAL DE LA UNI (EDUNI) SE UNEN EN ESTE ESFUERZO EDITORIAL PARA PONER A DISPOSICIÓN DE LA COMUNIDAD CIENTÍFICA LOS TRABAJOS DE LA NUEVA SERIE.""""

WE CONSIDER TWO INVERSE EIGENPROBLEMS FOR A REAL SYMMETRIC DOUBLY ARROWHEAD MATRIX A N (Q) , WHICH CONSIST OF CONSTRUCTING A N (Q) FROM TWO SPECIAL KINDS OF SPECTRA INFORMATION. THESE PROBLEMS WERE INTRODUCED IN [11], WHERE THE PRINCIPAL RESULTS ARE SUFFICIENT CONDITIONS FOR BOTH PROBLEMS TO HAVE A REAL SOLUTION. IN THIS PAPER, WE IMPROVE SUCH CONDITIONS, IN THE SENSE THAT ONE OF THE GIVEN CONDITIONS IMPLIES THE REST. THE RESULTS ARE CONSTRUCTIVE AND GENERATE ONE NUMERICAL PROCEDURE TO CONSTRUCT THE SOLUTION MATRIX A N (Q) .

WE OBTAIN A LOWER BOUND FOR THE ENTROPY OF A (NOT NECESSARILY INVARIANT) BOREL PROBABILITY MEASURE WITH RESPECT TO AN UPPER SEMICONTINUOUS SET-VALUED MAP AS THE PRODUCT OF THE LOWER DIMENSION OF THE MEASURE AND THE LOGARITHMIC EXPANSION RATE. THIS IS A GENERALIZATION OF THE WELL-KNOWN MEASURE-PRESERVING SINGLE-VALUED CASE.

WE OBTAIN A LOWER BOUND FOR THE ENTROPY OF A BOREL PROBABILITY MEASURE (NOT NECESSARILY INVARIANT) WITH RESPECT TO AN UPPER SEMICONTINUOUS SET-VALUED MAP AS THE PRODUCT OF THE LOWER DIMENSION OF THE MEASURE AND THE LOGARITHMIC EXPANSION RATE. THIS IS A GENERALIZATION OF THE WELL-KNOWN MEASURE-PRESERVING SINGLE-VALUED CASE.

IN THIS WORK A NEW TYPE OF MATRIX CALLED CIRCULANT-LIKE MATRIX IS INTRODUCED. THIS TYPE OF MATRIX INCLUDES THE CLASSICAL K-CIRCULANT MATRIX, INTRODUCED IN [4], IN A NATURAL SENSE. ITS EIGENVALUES AND ITS INVERSE AND SOME OTHER PROPERTIES ARE STUDIED, NAMELY, IT IS SHOWN THAT THE INVERSE OF A MATRIX OF THIS TYPE IS ALSO A MATRIX OF THIS TYPE AND THAT ITS FIRST ROW IS THE UNIQUE SOLUTION OF A CERTAIN SYSTEM OF LINEAR EQUATIONS. ADDITIONALLY, FOR SOME OF THESE MATRICES WHOSE ENTRIES ARE WRITTEN AS FUNCTION OF HORADAM, FIBONACCI, JACOBSTHAL AND PELL NUMBERS WE STUDY ITS EIGENVALUES AND WRITE IT AS FUNCTION OF THOSE NUMBERS. MOREOVER, THE SAME STUDY IS DONE IF THE ENTRIES ARE ARITHMETIC SEQUENCES.

THE GEOMETRIC LORENZ ATTRACTOR IS AN ATTRACTOR SET CONSTRUCTED IN SUCH A WAY THAT IT SATISFIES THE MAIN QUALITATIVE PROPERTIES EVIDENCED ON THE LORENZ SYSTEM EQUATIONS, PARTICULARLY THE FACT THAT THIS ATTRACTOR IS A ROBUSTLY TRANSITIVE SET. IN THIS PAPER WE PROVE THE -ROBUST TRANSITIVITY BY USING GEOMETRIC PROPERTIES FOR SINGULAR HYPERBOLIC SETS AND WITHOUT THE ASSUMPTION OF THE UNIFORMLY LINEARIZING COORDINATES AROUND THE SINGULARITY.

IN THIS PAPER, CLOSED FORMULAS FOR THE EIGENVECTORS OF A PARTICULAR CLASS OF MATRICES GENERATED BY GENERALIZED PERMUTATION MATRICES, NAMED GENERALIZED CIRCULANT MATRICES, ARE PRESENTED.

IN THIS PAPER WE PROVE SOME KIND OF STRUCTURAL STABILITY DEFINED AS USUAL BUT RESTRICTED TO A CERTAIN SUBSET OF ONE-DIMENSIONAL MAPS COMING FROM FIRST RETURN MAPS ASSOCIATED TO SINGULAR CYCLES FOR VECTOR FIELDS IN MANIFOLDS WITH BOUNDARY. THE MOTIVATION IS THE STABILITY OF THE SINGULAR HORSESHOES INTRODUCED BY LABARCA AND PACIFICO WHERE AN EXPANDING CONDITION ON THE SINGULARITY HOLDS. HERE WE OBTAIN ANALOGOUS RESULT BUT UNDER A CONTRACTING CONDITION.

WE STUDY A TYPE OF PERTURBED POLYNOMIAL HAMILTONIAN SYSTEM IN 1:1 RESONANCE. THE PERTURBATION CONSISTS OF A HOMOGENEOUS QUARTIC POTENTIAL INVARIANT BY ROTATIONS OF ?/2 RADIANS. THE EXISTENCE OF PERIODIC SOLUTIONS IS ESTABLISHED USING REDUCTION AND AVERAGING THEORIES. THE DIFFERENT TYPES OF PERIODIC SOLUTIONS, LINEAR STABILITY, AND BIFUR- CATION CURVES ARE CHARACTERIZED IN TERMS OF THE PARAMETERS. FINALLY, SOME CHOREOGRAPHY OF BIFURCATIONS ARE OBTAINED, SHOWING IN DETAIL THE EVOLUTION OF THE PHASE FLOW.

WE STUDY THE POLYNOMIAL ENTROPY OF HOMEOMORPHISMS ON COMPACT METRIC SPACES. WE CONSTRUCT A HOMEOMORPHISM ON A COMPACT METRIC SPACE WITH VANISHING POLYNOMIAL ENTROPY THAT IT IS NOT EQUICONTINUOUS. ALSO WE GIVE EXAMPLES WITH ARBITRARILY SMALL POLYNOMIAL ENTROPY. FINALLY, WE SHOW THAT EXPANSIVE HOMEOMORPHISMS AND POSITIVELY EXPANSIVE MAPS OF COMPACT METRIC SPACES WITH INFINITELY MANY POINTS HAVE POLYNOMIAL ENTROPY GREATER THAN OR EQUAL TO 1.